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KAN'.A", i,II f Mt) IMiHUl tWHAHY I 0 ODD1 DSOSQEb .^*— . * A HISTORY OF MECHANICS A HISTORY MECHANICS II V HKNft IHK;AS *>*; «u*Nr£tt*;N(;»:H <»r tin nun,*1, i*t»ti FOKKWOttl* in IXM IS Hi-; J, K. M\UI)0\ w f;mi*t4iNf ^M^ifATKr-swtt/ ^\, tN<:,, NKW Y FOREWORD The history of mechanics is one of the most important branches of the history of science. From earliest times man has sought to develop tools that would enable him to add to his power of action or to defend himself against the dangers threatening him. Thus he was uncons-* ciously led to consider the problems of mechanics. So we see the first scholars of ancient times thinking about these problems and arriving more or less successfully at a solution. The motion of the stars which, from the Chaldean shepherds to the great Greek and Hellenistic astronomers, was one of the first preoccupations of human thought, led to the discovery of the true laws of dynamics. As is well known, although the principles of statics had been correctly presented by the old scholars those of dynamics, obscured by the false conceptions of the aristotelian school, did not begin to see light until the end of the Middle Ages and the beginning of the modern era. Then came the rapid development of mechanics due to the memor* able work of Kepler, Galileo, Descartes, Huyghens and Newton; the codification of its laws by such men as Euler, Lagrange and Laplace ; and the tremendous development of its various branches and the endlessly increasing number of applications in the Nineteenth and Twentieth Centuries. The principles of mechanics ivere brought to such a high degree of perfection that fifty years ago it was believed that their develop ment was practically complete. It was then that there appeared^ in sue- cession^ two very unexpected developments of classical mechanics— on the one hand, relativistic mechanics and on the, other, quantum and wave mechanics. These originated in the necessity of interpreting the very delicate phenomena of electromagnetism or of explaining the observable processes on the atomic scale. Whereas reativistic mechanics, while up setting our usual notions of time and space only*, in a sense, completed and crowned the work of classical mechanics^ the quantum and wave mechanics brought us more radically new ideas and forced us to give up the continuiuty and absolute determinism of elementary phenomena. 8 HISTORY OF MECHANICS Relativistic and quantum mechanics today form the two highest peaks of the progress of our knowledge in the whole field of mechanical phenomena. To appraise the evolution of mechanics from its origin up to the present time would be obviously a difficult task demanding a considerable amount of work and thought. Few men would be tempted to write such a history of mechanics ; for its compilation would require not only a wide and thorough knowledge of all the branches of mechanics ancient and modern, but also a great patience, a well-informed scholarship and an acute and critical mind. These varied qualities M. Rene Dugas — who has already become known for his fine studies on certain particular themes in the history of dynamics and for his critical essays on different matters in class ical, relativistic and quantum mechanics — unites these to a high degree. More than this, he has tackled this overwhelming task and, after several years, has brought it to a successful conclusion. The important work that he now publishes on the history of mechanics constitutes a compre hensive view of the greatest interest which will be highly appreciated by all those who study the history of scientific thought. Mr. Dugas' book is in certain ways comparable with Ernest Mack's famous book " Mechanics, A historical and critical presentation of its development. " Certainly the reading of Mach's book, so full of original ideas and profound comments, is still extremely instructive and absorbing. But Mr. Dugas'' history of mechanics has the advantage of being less systematic and more complete. Mactfs thought was in fact dominated by the general ideas which secured his adherence in Physics to the energetic school and in Philosophy to the positivistic thesis. He frequently sought to find an illustration, in the history of mechanics, of his own ideas. Often this gives his book a character which is a little too systematic, that of a thesis in which the arguments in favour of preconceived ideas are rehearsed. Mr. Dugas' attitude is quite different. A scrupulous historian, he has patiently followed all the vagaries of thought of the great students of the subject, collating their texts carefully and always preserving the strictest objectivity. More impartial than Mach, Mr. Dugas has been helped by the, deve lopment of historical criticism on the one hand, by the progress of science on the other, and has been able to be more complete. He has given us a much more detailed picture of the efforts that were made and the remits obtained in Antiquity and, especially, in the Middle Ages. It is parti cularly to the authoritative ivork of Pierre Duhem that Mr. Dugas oiws FOREWORD 9 his ability to show us the important contributions made to the development of the principles of mechanics by masters like Jordanus of IVemore, Jean Buridan, Albert of Saxony , Nicole Oresme and a great artist of universal interest like Leonardo da Vinci. Of DuhemSs magnificent researches — which are often a little difficult to study in that eminent and erudite phy- sicisfs original text, usually lengthy and somewhat vague — Mr. Dugas has been able to make, in a few pages, a short presentation that the reader will read easily and tvith the greatest profit. Well informed of the most recent progress of the science, the author, accustomed to reflect of the new contemporary forms of mechanics, has devoted the last part of his book to relativistic mechanics and wave and quantum mechanics. This very accurate presentation made by following closely, as is the authors practice, the ideas of the innovators and the text of their writings, naturally makes Mr. Dugas' history of mechanics much more complete that those of his predecessors. The central part of the book, devoted to the developments of mechanics in the Seventeenth, Eighteenth and Nineteenth Centuries, has demanded a great amount of work^ for the material is immense. Being unable to follow all the details of the development of mechanics in the Eighteenth Century, and especially in the Nineteenth Century, Mr. Dugas has selected for a thorough study certain questions of special importance, either in themselves or because of the extensions which they hav& had into the con* temporary period. It seems to me that this selection has been made very skillfully and has enabled the author, without losing himself in details, to outline the principal paths followed by scientific thought in this domain. Perhaps, in, reading Mr. Dugas'' so clear text, the reader will not appreciate the work that the writing of such a book represents. Not only has Mr. Dugas had to sift various questions to select those which would most clearly illustrate the decisive turning-points in the progress of mechanics, but he has always referred to the original texts themselves, never wanting to accomplish the task at second hand. When^ for ex« ample ^ he summarises for us the work of Kepler in a few p&g®s>> it is after having re-examined and* in some way, rethought these arguments —often complicated and a tittle quaint and, moreover, written in a bad lAitin whose meaning is often difficult to appreciate—which enabled the great astronomer to discover the correct laws of the motion of the planets. It is this necessary conjunction of the procedures of a patient erudition and a wide knowledge of the past and present results of the science which mak$s 10 HISTORY OF MECHANICS the history of science particularly difficult and restricts the number of those who can, with profit, devote themselves to it. Therefore Mr. Rene Dugas should be warmly thanked for having placed at the service of the history of science, qualities of mind and methods of work rarely united in one man, and for having given us a remarkable work which will remain a document of the first rank for the historian of mechanics. of the AcadAmie franyaite permanent secretary of the Academic des sciences , PREFACE Mechanics is one of the branches of physics in "which the number of principles is at once very few and very rich in useful consequences. On the other hand, there are few sciences which have required so much thought — the conquest of a few axioms has taken more than 2000 years. As Mr. Joseph Per^s has remarked, to speak of the miracle of Greece or of the night of the Middle Ages in the evolution of mechanics is not possible. Correctly speaking, Archimedes was able to conquer statics and knew how to construct a rational science in which the precise deductions of mathematical analysis played a part. But hellenic dynamics is now seen to be quite erroneous. It was however, in touch with every-day observation. But, being unable to recognise the func tion of passive resistances and lacking a precise kinematics of accelerated motion, it could not serve as a foundation for classical mechanics. The prejudices of the Schoolmen, whose authority in other fields was undisputed, restricted the progress of mechanics for a long period. Annotating Aristotle wan the essential purpose of teaching throughout the Middle Ages. Not that the mediaeval scholars lacked originality. Indeed, they displayed an acute subtlety which haa never been surpassed, but most often they neglected to take account of observation, preferring to exercise their minds in a pure field. Only the astronomers were an exception and accumulated the facts on which, much later, mechanics was to be based* The Thirteenth Century had, however, an original school of ntatics which advocated, in the treatment of heavy bodies, a new principle — under the name of gmvitax secundum situm— that was to develop into the principle of virtual work; moreover, thitf principle solved, long before Stevin and Galileo, the problem of tin* equilibrium, of a heavy body on an inclined plane, which Pappus had not sueeeecled in doing correctly. In the fourteenth century, Buridan formulated the first theory of energy under the name of impetus. Thin theory explicitly departs from the Peripatetic ideas, which demanded the constant inter vention of a mover to maintain violent motion in the Aristotelian sense. Incorporated into a continued tradition in which it was deformed in 12 HISTORY OF MECHANICS order to conform to an animist doctrine, which in the hands of the German metaphysicians of the fifteenth century was to subsist with Kepler, the theory of impetus became, in the hands of Benedetti, an early form of the principle of inertia, while one of its other aspects was to become, after a long polemic, the doctrine of vis viva. And in the Fourteenth Century, the Oxford School, which in other respects indulged in such artificial quibbling, was to clarify the laws of the kinematics of uniformly accelerated motion. The mechanics of the Middle Ages received something of a check during the Renaissance, which caused a return to classical traditions. The Schools were attacked by the humanists. Yet, before Galileo, Dominico Soto successfully formulated the exact laws of heavy bodies even if he did not verify them experimentally. Under what may seem an ambitious title, A History of Mechanics, we shall deal with the evolution of the principles of general mechanics, while we shall omit the practical applications and, a fortiori, tech nology. As far as possible we shall follow a chronological order, in the manner of elementary text books which begin with early history and end with the latest war. After considerable reflexion, this order has seemed to us preferable to the one adopted by important critics, which consists in choosing a given principle or problem and analysing its evolution throughout the centuries. This latter method offers the advantage of isolating a theory and treating it closely ; it lends itself to short cuts which are striking but which sometimes seem a little artificial : the different key problems of mechanical science evolved in fact along parallel lines, profiting by the progress made in mathematical language ; what is more, these problems were interconnected. We have preferred here to follow the elementary order in time. Each century will thus appear in full light, with its own mentality and atmosphere. So we jump necessarily from one theme to another, but the works find once more their unity and their natural background, without the distortion caused by juxtaposing them in time. The present book is divided into five parts. The first treats of the originators, the precursors, from the beginning up to and including the Sixteenth Century; the second, of the formation of classical mechanics. In this domain the Seventeenth Century deserves to be considered as the great creative century, with three great peaks formed by Galileo, Huyghens and Newton. The third part is devoted to the Eighteenth Century, which emerges as the century of the organisation of mechanics and finds its climax in the work of Lagrange, immediately PREFACE 13 preceded by Euler and d'Alembert. The development of mathematical analysis enabled mechanics to take a form which, for a considerable time, appeared to be finally established, and which is still a part of the classical teaching. We have found ourselves somewhat embarrassed in the writing of the fourth part, which is concerned with classical mechanics after Lagrange. Indeed, nothing would be gained by duplicating the textbooks. Therefore we have confined ourselves to a selection from the work of the Nineteenth Century and the beginning of the Twentieth Century. This selection may appear somewhat arbitrary to the well- informed reader. Moreover, it is rather interesting to observe that the uneasiness about the structure of classical mechanics which is evident in the writings of the critics did not prepare the way for the relativistic and quantum revolutions, dated 1905 and 1923 respectively. These came from outside, imposed by the necessities of optics, electro- magnetism and the theory of radiation. However, these reflections of the critics were not valueless, for they showed that the lagrangian science was not intangible and that the axioms of mechanics, to use Mach's words, " not only assumed but also demanded the continual control of experiment." Thus the determinism — or, if it is preferred, legality — of which the students of mechanics were so proud and which made their science the model of all physical theories, now appears, after the development of wave mechanics, as justified in the macroscopic domain because of statistical compensation between the individuals of a large assembly, without there being an underlying determinism. The fact of collecting in one book the stammerings of the early students, the creation and organisation of the classical science and the rudiments of the new mechanics— the object of the fifth part of this book — may appear a wager. It is only so iu appearance. Indeed, the original works never had that codified aspect which is, of necessity, lent them by the textbooks. Just as the French currency remained stable for more than a century, Lagrange'n mechanics was able to appear as complete for a period of roughly the same length* Despite its mathematical perfection, it had no other foundation that that of common experiment. The double revolution of 1905 and 1923— the second much more radical than the first—profoundly disturbed the classical structure. For these new doctrines intended to rule over the whole of physics, only admitting the validity of classical mechanics in the limits at which the velocity oflight can be considered infinite and Planck's universal constant negligible. As regards the principles, cer tain thinkers have made the mistake of incorporating into a system 14 HISTORY OF MECHANICS what was only a stability of fact, a stage in evolution, as long as this stability appeared to be verified and however important were, and still are, its conquests. For the historian, who is only a spectator and who, by his profession, is aware of the fragility of our anthropomorphic science, these revolu tions are not an object of scandal or even a surprise. This does not mean that innovations must be accepted without analysis, or new dogmas professed without reserve. It is difficult for us to indicate in detail what this work owes to the great historians and critics of mechanics. From Duhem we have taken what material he could extract from the manuscripts of the Middle Ages, at the same time bringing to this semi-darkness the light of his particularly profound and alert mind. We must confess however that we have had to disagree with several of his opinions, which appeared too categorical to us. Duhem had undergone the polarisation of the investigator which leads him to attach too great an importance some times to the original he has just discovered. Besides, Duhem's attach ment to energetics made him somewhat biassed. For example, the principle of virtual work triumphed over that of virtual velocity which, up to the the early writings of Galileo, remained inspired by peripatetic doctrines, but it already explained correctly the laws of the equilibrium of simple machines, and long preceded the former. Curiously enough, Duhem makes Leonardo da Vinci the centre of his studies of the Pre cursors, to the extent of considering the Unknown Man of the Thirteenth Century, that same one who discovered the law of the equilibrium of a heavy body on an inclined plane, as a precursor of Leonardo, whereas this latter was always in error on this problem, and of qualifying as plagiaries or successors of Leonardo a large number of authors of the Sixteenth Century who may very well not have come under the influence of the great painter who in mechanics was scarcely more than an amateur of genius. These few reserves, for which we beg to be excused, do not prevent Duhem's historical work from being of the greatest importance. An indefatigable reader, he succeeded not only in bringing to life workfi of the Scholastics of the Middle Ages, that were until then little known, but in establishing between them and the classical period filiations of indisputable interest. It is certain that, on more than one point, this Scholastic sheds light on and prefigures Descartes. From Emile Jouguet, who honoured us with his teaching, we have borrowed several of his Lectures — given conscientiously and with great regard for the original authors— and a number of opinions that were, PREFACE 15 out of modesty, consigned to notes, lest they should hide his perfect knowledge of the Ancients. In many places we have cited the very personal, and sometimes very judicious, observations of Mach. His Mechanics was one of the first systematic works of its kind and represented both a very wide reading and a critical mind of remarkable independence.1 Strictly speaking, Painleve did not treat the history of mechanics. On his own account, with the analytical mind that he applied to every thing, he rethought the evolution of mechanics. The lectures which he gave us at the £cole pofytechnique were revised and developed in his Axiomes de la Mecanique. This contains not only an original discussion of the classical principles, but an often constructive and always valuable criticism of relativistic doctrines. In spite of the contributions of the great critics we have just mention ed, in spite of the several researches of the original authors themselves which this book contains, we do not conceal its imperfections and its omissions. Certain of these omissions, especially from the classical field after Lagrange, have been accepted deliberately ; others may be unknown to us and, for this reason, more serious. We have not sought to restrict ourselves too narrowly to our nubject, and have made some incursions into the domain of astronomy and that of hydrodynamics when it seemed that these served our purpose. But a presentation of a system of the world, or a complete history of the mechanics of fluids, should not be looked for here ; these subjects themselves would require whole volumes. This book will only be read with profit by those who already have some knowledge of the didactic aspect of things. It also presupposes, as does all mechanics, a somewhat extensive mathematical background. Our purpose will have been achieved if the reader finds in it, with less effort than it has cost us to unite and explain the original texts, a reflec tion of the joy of knowledge that we have found. I must thank the " Editions du Griffon " for having applied all their recourses to the production of this book, 1 Throughout this book, Du HEM'S Origines de ia Statique are indicated by the initials 0. 6'., MACK'S Mechanics by the imtial M., and JOUGUKT'B Lectures de Mtca- nique by the initials L. M. PART ONE THE ORIGINS CHAPTER ONE HELLENIC SCIENCE 1. ARISTOTELIAN MECHANICS. For lack of more ancient records, history of mechanics starts with Aristotle (384-322 B. C.) or, more accurately, with the author of the probably apocryphal treatise called Problems of Mechanics (Mriywvmv. nQoftkrifjiVLTy.) . This is, in fact, a text-book of practical mechanics devoted to the study of simple machines. In this treatise the power of the agency that sets a body in motion is defined as the product of the weight or the mass of the body — the Ancients always confused these concepts — and the velocity of the motion which the body acquires. This law makes it possible to formulate the condition of equilibrium of a straight lever with two unequal arms which carry unequal weights at their ends. Indeed, when the lever rotates the velocities of the weights will be proportional to the lengths of their supporting arms, for in these circumstances the powers of the two opposing powers cancel each other out. The author regards the efficacy of the lever as a consequence of a magical property of the circle. " Someone who would not be able to move a load without a lever can displace it easily when he applies a lever to the weight. Now the root cause of all such phenomena is the circle. And this is natural, for it is in no way strange that something remarkable should result from something which is more remarkable, and the most remarkable fact i« the combination of oppoeites with each other. A circle is made up of such opposites, for to begin with it is made up of something which moves and something which remains stationary. ..." * In this way Problems of Mechanics reduces the study of all simple machines to one and the same principle. ** The properties of the balance are related to those of the circle and the properties of the lever to those of the balance. Ultimately most of the motions in mechanics are related to the properties of a lever. '* 20 THE ORIGINS To Aristotle himself, just as much in his Treatise on the Heavens, (IIsQi OVQWOV) as in his Physics, concepts belonging to mechanics were not differentiated from concepts having a more general significance. Thus the notion of movement included both changes of position and changes of kind, of physical or chemical state. Aristotle's law of powers, which he caUed Mvayug or fcyj^g, is formulated in Chapter V of Book VII of his Physics in the following way. " Let the motive agency be a, the moving body /?, the distance tra velled y and the time taken by the displacement be <5. Then an equal power, namely the power a, will move half of /? along a path twice y in the same time, or it will move it through the distance y in half the time d. For in this way the proportions will be maintained. " Aristotle imposed a simple restriction on the application of this rule— a small power should not be able to move too heavy a body, " for then one man alone would be sufficient to set a ship in motion. " This same law of powers reappears in Book III of the Treatise on the Heavens. Its application to statics may be regarded as the origin of the principle of virtual velocities which will be encountered much later. In another place Aristotle made a distinction between natural motions and violent motions. The fall of heavy bodies, for example, is a natural motion, while the motion of a projectile is a violent one. To each thing corresponds a natural place. In this place its substantial form achieves perfection — it is disposed in such a way that it is subject as completely as possible to influences which are favourable, and so that it avoids those which are inimical. If something is moved from its natural place it tends to return there, for everything tends to perfection. If it already occupies its natural place it remains there at rest and can only be torn away by violence. In a precise way, for Aristotle, the position of a body is the internal surface of the bodies which surround it. To his most faithful commen tators, the natural place of the earth is the concave surface which deiines the bottom of the sea, joined in part to the lower surface of the atmo sphere, the natural place of the air.1 Concerning the natural motion of falling bodies, Aristotle maintained in Book I of the Treatise on the Heavens that the " relation which weights have to each other is reproduced inversely in their durations of fall. If a weight falls from a certain height in so much time, a weight which is twice as great will fall from the same height in half the time* In his Physics (Part V), Aristotle remarked on the acceleration of 1 Cf. DXIHEM, Origines de la Statique, Vol. II, p. 21. Throughout the prenent hook this work of Duhem will be indicated by the letters 0. S. HELLENIC SCIENCE 21 falling heavy bodies. A body is attracted towards its natural place by means of its heaviness. The closer the body comes to the ground, the more that property increases. If the natural place of heavy bodies is the centre of the World, the na tural place of light bodies is the region contiguous with the Sphere of the Moon. Heavenly bodies are not subject to the laws applicable to terres trial ones — every star is a body as it were divine, moved by its own divinity. We return to terrestrial mechanics. All violent motion is essentially impermanent. This is one of the axioms which the Schoolmen were to repeat — Nullum violentum potest esse perpetuum. Once a projectile is thrown, the motive agency which assures the continuity of the motion resides in the air which has been set in motion. Aristotle then assumes that, in contrast to solid bodies, air spontaneously preserves the impul sion which it receives when the projectile is thrown, and that it can in consequence act as the motive agency during the projectile's flight. This opinion may seem all the more paradoxical in view of the fact that Aristotle remarked, elsewhere, on the resistance of the medium. This resistance increases in direct proportion to the density of the me dium. " If air is twice as tenuous as water, the same moving body will spend twice as much time in travelling a certain path in water as in travelling the same path in air. " Aristotle also concerned himself with the composition of motions. " Let a moving body be simultaneously actuated by two motions that are such that the distances travelled in the same time are in a constant proportion. Then it will move along the diagonal of a parallelogram which has as sides two lines whose lengths are in this constant relation to each other. " On the other hand, if the ratio between the two com ponent distances travelled by the moving body in the name time varies from one instant to another, the body cannot have a rectilinear motion. " In such a way a curved path is generated when the moving body is animated by two motions whose proportion does not remain constant from one instant to another. " These propositions relate to what we now call kinematics. But Aristotle immediately inferred from them dynamical results concerning the composition of forces. The connection between the two disciplines is not given, but as Dutxem has indicated, it is easily supplied by making use of the law of powers— a fundamental principle of aristotelian dyna mics. In particular, let us consider a heavy moving body describing some curve in a vertical plane. It is clear that the body is actuated by two motions simultaneously. Of these, one produces a vertical descent while the other, according to the position of the body on its trajectory, results in an increase or a decrease of the distance from the 22 THE ORIGINS centre. In Aristotle's sense, the body will have a natural falling motion due to gravity, and will be carried horizontally in a violent motion. Consider different moving bodies unequally distant from the centre of a circle and on the same radius. Let this radius, in falling, rotate about the centre. Then it may be inferred that for each body the relation of the natural to the violent motion remains the same. " The contem plation of this ecpiality held Aristotle's attention for a long time. He appears to have seen in it a somewhat mysterious correlation with the law of the equilibrium of levers. " x Aristotle believed in the impossibility of a vacuum (Physics, Book IV, Chapter XI) on the grounds that, in a vacuum, no natural motion, that is to say no tendency towards a natural place, would be possible. Incidentally this idea led him to formulate a principle analogous to that of inertia, and to justify this in the same way as that used by the great physicists of the XVIIIth Century. " It is impossible to say why a body that has been set in motion in a vacuum should ever come to rest ; why, indeed, it should come to rest at one place rather than at another. As a consequence, it will either necessarily stay at rest or, if in motion, will move indefinitely unless some obstacle comes into collision with it. " Aristotle's ideas on gravitation and the figure of the Earth merit our attention, if only because of the influence which they have had on the development of the principles of mechanics. First we shall quote from the Treatise on the Heavens (Book II, Chapter XIV). " Since the centres of the Universe and of the Earth coincide, one should ask one self towards which of these heavy bodies and even the parts of the Earth are attracted. Are they attracted towards this point because it is the centre of the Universe or because it is the centre of the Earth ? It is the centre of the Universe towards which they must be attracted. . . . Consequently heavy bodies are attracted towards the centre of the Earth, but only fortuitously, because this centre is at the centre of the Universe. " If the Earth is spherical and at the centre of the World, what happens if a large weight is added to one of the hemispheres ? The answer to this question is the following. " The Earth will necessarily move until it surrounds the centre of the World in a uniform way, tlie tendencies to motion of the different parts counterbalancing one an other." Duhem points out that the centre, TO ueaov, that in every body is attracted to the centre of the Universe, was not defined in a precise 1 DUHEM, 0. S., Vol. I, p. 110. Note here that,/or the same fall, the longer the'lever the less the natural motion will be disturbed. It is therefore natural to assume that a weight has more power at the end of a long lever than at the end of a short one. HELLENIC SCIENCE 23 way by Aristotle. In particular, Aristotle did not identify it with the centre of gravity, of which he was ignorant.1 In this same treatise Aristotle repeatedly enumerates the arguments for the spherical figure of the Earth. He distinguishes a posteriori arguments, such as the shape of the Earth's shadow in eclipses of the Moon, the appearance and disappearance of constellations to a traveller going from north to south, from a priori arguments, of which he says — " Suppose that the Earth is no longer a single mass, but that, poten tially, its different parts are separated from each other and are placed in all directions and attracted similarity towards the centre. Then let the parts of the Earth which have been separated from each other and taken to the ends of the World be allowed to reunite at the centre ; let the Earth be formed by a different procedure — the result will be exactly the same. If the parts are taken to the ends of the World and are taken there similarily in all directions, they will necessarily form a mass which is symmetrical. Because there will result an addition of parts which are equal in all directions, and the surface which defines the mass produced will be everywhere equidistant from the centre. Such a surface will therefore be a sphere. But the explanation of the shape of the Earth would not be changed in any way if the parts which form it were not taken in equal quantities in all directions. In fact, a larger part will necessarily push away a smaller one which it finds in front of it, for both have a tendency towards the centre and more powerful weights are able to displace lesser ones. " To Aristotle, heaviness does not prove rigorously that the Earth will be spherical, but only that it will tend to be so. On the other hand, for the surface of water, this is obvious if it is admitted that " it is a property of water to run towards the lowest places, " that is, towards places which are nearest the centre. Let ftey be an arc of a circle with centre a ; the line a<5 is the shortest distance from cc to (iy* " Water will run towards <3 from all Hides until its surface becomes equidistant from the centre. It therefore follows that the water takes up the same length on all the lines radiating from the Fig. 1 centre. It then remains in equilibrium. But the locus of equal lines radiating from a centre in a circumference of a circle. The surface of the water, /5tey, will therefore be spherical. " Adrastus (360-317 B. C), commenting on Aristotle, made the pre- * DUHEM, 0. 5., Vol. II, p. II. 24 THE ORIGINS ceding proof precise in the following terms. " Water will run towards the point <5 until this point, surrounded by new water, is as far from oc as /3 and y. Similarity, all points on the surface of the water will be at an equal distance from a. Therefore the water exhibits a spherical form and the whole mass of water and Earth is spherical. " Adrastus supplemented this proof with the following evidence, which was destined to become classical.1 " Often, during a voyage, one cannot see the Earth or an approach ing ship from the deck, while sailors who climb to the top of a mast can see these things because they are much higher and thus overcome the convexity of the sea which is an obstacle. " We shall say no more about aristotelian mechanics. However inad equate they may seem now, these intuitive theories have their origin in the most everyday observations, precisely because they take the passive resistances to motion into account. To an unsophisticated observer, a horse pulling a cart seems to behave according to the law of powers, in the sense that it develops an effort which increases regularly with the speed. In order to break away from Aristotle's ideas and to con struct the now classical mechanics, it is necessary to disregard the va rious ways in which motion may be damped, and to introduce these explicitly at a later stage as factional forces and as resistances of the medium. However it may be, aristotelian doctrines provided the fabric of thought in mechanics for nearly two thousand years, so that even Galileo, who was to become the creator of modern dynamics, made his first steps in science in commenting Aristotle, and proved in his early writings to be a faithful Peripatetic ; which, it may be said in passing, in no way diminishes his glory as a reformer, on the contrary, it only adds to it. 2. THE STATICS OF ARCHIMEDES. Unlike Aristotle, whose mechanics is integrated into a theory of physics which goes so far as to incorporate a system of the world, Archi medes (287-212 B.C.) made of statics an autonomous theoretical science, based on postulates of experimental origin and afterwards supported by mathematically rigourous demonstrations, at least in appearance* Here we shall follow the treatise On the Equilibrium of Planes or on the Centres of Gravity of Planes 2 in which Archimedes discussed the principle of the lever. X ™s thesis of ADRASTUS is known to us by means of A Collection of Mathematical Knowledge useful for the Reading of Plato, by THEON OF SMYRNA. 2 Translation by PEYRARD, Paris, 1807. The reader .should also refer to that of P. VER EECKE, Paris and Anvers, Desclee de Brouwer, 1938. HELLENIC SCIENCE 25 Archimedes made the following postulates as axioms — 1) Equal weights suspended at equal distances (from a fulcrum) are in equilibrium. 2) Equal weights suspended at unequal distances cannot be in equilibrium. The lever will be inclined towards the greater weight. 3) If weights suspended at certain distances are in equilibrium, and something is added to one of them, they will no longer be in equilibrium. The lever will be inclined towards the weight which has been increased. 4) Similarily, if something is taken away from one of the weights, they will no longer be in equilibrium, but will be inclined towards the weight which has not been decreased. 5) If equal and similar plane figures coincide, their centres of gravity will also coincide. (The concept of Centre of Gravity appears to have been defined by Archimedes in an earlier manuscript, of which no trace remains.) 6) The centres of gravity of unequal but similar figures are similarity placed. 7) If magnitudes suspended at certain distances arc in. equilibrium, equivalent magnitudes suspended at the same distances will also be in equilibrium. 8) The centre of gravity of a figure which is nowhere concave is necessarily inside the figure. With this foundation, Archimedes proved the following propositions. Proposition I. — When weights suspended at equal distances are in equilibrium, these weights are equal to each other. (Proof by reductio ad absurdum based on Postulate 4).) Proposition II. — Unequal weights suspended at equal distances will not be in equilibrium, bxit the greater weight will fall. (Proof based on Postulates 1) and 3).) Proposition III. — Unequal weights suspended at unequal distances may be in equilibrium, in which case the greater weight will be suspended at the shorter distance. (Proof based on Postulates 4), 1) and 2). Thin proof only confirms the second part of the proposition, and does not demonstrate the possi- bility of the equilibrium of two unequal weights. This must be regarded as an additional postulate of experimental origin.) 26 THE ORIGINS Proposition IV. — If two equal magnitudes do not have the same centre of gravity, the centre of gravity of the magnitude made up of these two magnitudes is the point situated at the middle of the line which joins their centres of gravity. (The proof, based on Postulate 2), is a demonstration by reductio ad absurdum which, moreover, assumes that the centre of gravity of the combined magnitude lies on the line joining the centres of gravity of the component magnitudes.) Proposition V. — If the centres of gravity of three magnitudes lie on the same straight line, and if the magnitudes are equally heavy and the distances between their centres of gravity are equal, the centre of gravity of the combined magnitude will be the point which is the centre of gravity of the central magnitude. (This is a corollary of Proposition IV, which Archimedes later extended to the case of n magnitudes. The enunciation is suitably modified if n is even.) Proposition VI. — Commensurable magnitudes are in equilibrium when they are reciprocally proportional to the distances at which they are suspended. A 2 8 L E C H D K f 1 1 1 1 1 Fig. 2 " Let the commensurable magnitudes be A and B, and let their centres of gravity be the points A and B. Let ED be a certain length and suppose that the magnitude A is to the magnitude B as the length DC is to the length CE. It is necessary to prove that the centre of gravity of the magnitude formed of the two magnitudes A and B i« the point C. " Since A is to B as DC is to CE and since the areas A and B are commensurable, the lengths DC and CE will also be commensurable. Therefore the lengths EC and CD have a common measure, say IV. HELLENIC SCIENCE 27 Suppose that each of the lengths DH and DK is equal to the length EC, and that the length EL is equal to the length DC. Since the length DH is equal to the length CJ5, the length DC will be equal to the length .EH, and the length LE will be equal to the length EH. Hence the length LH is twice the length DC, and the length HK twice the length CE. Therefore the length N is a common measure of the lengths DH and HK since it is a common measure of their halves. But A is to B as DC is to CE, so that A is to B as LH is to HK. Let A be as many times greater than Z as LH is greater than N. The length LH will be to the length IV as A is to Z. But KH is to LH as B is to A. Therefore, by equality, the length KH is to the length N as B is to Z. Then B is as many times greater than Z as KH is a multiple of N. But it has been arranged that A is also a multiple of Z, Therefore Z is a common measure of A and B. Consequently, if LH is divided into segments each equal to JV, and A into segments each equal to Z, A will contain as many segments equal to Z as LH contains segments equal to N. Therefore, if a magnitude equal to Z is applied to each segment of LH in such a way that its centre of gravity is at the centre of the segment, all the magnitudes1 will be equal to A. Further, the centre of gravity of the magnitude made up of all these magnitudes will be the point JS, remembering that they are an even number and that LE is equal to HE (Proposition V). Similarity it could be shown that if a magnitude equal to Z was applied to each of the segments of KH> with its centre of gravity at the centre of each segment, all those magnitudes l would be equal to B and that the combined centre of gravity would be D. But the magnitude A is applied at the point E and the magnitude B at the point D. Therefore certain equal magnitudes are placed on the same line, their centres of gravity are separated from each other by the same interval and they are an even number. It is therefore clear that the centre of gravity of the magnitude composed of all these magnitudes is the point at the middle of the line on which the centres of gravity of the central magnitudes lie (Proposition V). But the length LJE is equal to the length CD and the length EC to the length CK. Thus the centre of gravity of the magnitude composed of all these areas is the point C, Therefore, if the magnitude A is applied to the point E and the magnitude B to the point D, the two areas will be in equilibrium about the point C. " Archimedes then extended this proportion to the caae of magnitudes A and B which were incommensurable. This demonstration depended on the method of exhaustion. We have reproduced this proof of 1 Read, ** the combination of ali these magnitudes. " 28 THE ORIGINS Proposition VI in its entirety in order to illustrate the nature of Archi medes' logical apparatus. This should not be allowed, however, to create too great an illusion of power. Indeed, Archimedes assumes in this proof that the load on the fulcrum of a lever is equal to the sum of the two weights which it supports.1 Further, he made use of the principle of superposition of equilibrium states, without emphasising that this was an experi mental postulate. Finally, and this is the most telling objection to the proceeding analysis, Archimedes, together with those of his suc cessors who tried to improve his proof, tacitly made the hypothesis that the product PL measures the effect of a weight P placed at a distance L from a horizontal axis. In fact, in the case of complete symmetry which is envisaged in Archimedes' first postulate, equili brium obtains whatever law of the form Pf(L)is taken as a measure of the effect of the weight P. It is therefore impossible, with the help of Archimedes' postulates alone, to substantiate Proposition VI in a logical way.2 For the rest, the treatise On the Equilibrium of Planes is concerned with the determination of the centres of gravity of particular geome trical figures. After having obtained the centres of gravity of a tri angle, a parallelogram and a trapezium, Archimedes determined the centre of gravity of a segment of a parabola by means of an analysis which is a milestone in the history of mathematics (Book II, Pro position VIII). We shall now concern ourselves with Archimedes' treatise on Floating Bodies. The author starts from the following hypothesis— 66 The nature of a fluid is such that if its parts are equivalently placed and continuous with each other, that which is the least compress ed is driven along by that which is the more compressed. Each part of the fluid is compressed by the fluid which is above it in a vertical direction, whether the fluid is falling somewhere or whether it in driven from one place to another. " From this starting point, the following propositions derive in a logical sequence. Proposition I. — If a surface is intersected by a plane which always* passes through the same point and if the section is a circumference (of a circle) having this fixed point as its centre, the surface in that of a sphere. 1 This is a point which can be established rigorously by considerations of symmetry alone, as FOURIER was to show, much later, in his perfection of a similar attempt due to D'ALEMBERT. 2 Cf. MACH, Mechanics, p. 21. Throughout this work, Mach's treatise will be indicated by the letter M. HELLENIC SCIENCE 29 Proposition II. — The surface of any fluid at rest is spherical and the centre of this surface is the same as the centre of the Earth. This result had already, as we have seen, been enunciated by Aristotle. Proposition III. — If a body whose weight is equal to that of the same volume of a fluid (a) is immersed in that fluid, it will sink until no part of it remains above the surface, but will not descend further. We shall reproduce the proof of this proposition, which is the origin of Archimedes' Principle. " Let a body have the same heaviness as a liquid. If this is possi ble, suppose that the body is placed in the fluid with part of it above the surface. Let the fluid be at rest. Suppose that a plane which passes through the centre of the Earth intersects the fluid and the f. body immersed in it in such a way that the section of the fluid is Alt(!D and the section of the body is ERTF. Let K be the centre of the Earth, BHTC be the part of the body which is immersed in the fluid and DEFC the part which projects out of it. Construct a pyramid whose base is a parallelogram in the surface of the fluid (a) and whose apex is the centre of the Earth. Let the intersection of the faces of the pyramid by the plane containing the are ABCD be KL and KM. In the fluid, and below EFTII draw another spherical surface XOP having the point K as its centre, in such a way that XOP is the section of the surface by the plane containing the are ABCD. Take another pyramid equal to the first, with which it is contiguous and continuous, and such that the sections of its faces are KM and KN. Suppose that there is, in the fluid, another solid RSQY which is made of the fluid and is equal and similar to BHTC, that part of the body 30 THE ORIGINS EHTF which, is immersed in the fluid. The portions of the fluid which are contained by the surface XO in the first pyramid and the surface OP in the second pyramid are equally placed and continuous with each other. But they are not equally compressed. For the portions of the fluid contained in XO are compressed by the body EHTF and also by the fluid contained by the surfaces LM, XO and those of the pyramid. The parts contained in PO are compressed by the solid RSQY and by the fluid contained by the surfaces OP, MN and those of the pyramid. But the weight of the fluid contained between MN and OP is less than the combined heaviness of the fluid between LM and XO and the solid. For the solid RSQY is smaller than the solid EHTF— RSQY is equal to BHTC— and it has been assumed that the body immersed has, volume for volume, the same heaviness as the fluid. If therefore one takes away the parts which are equal to each other, the remainder will be unequal. Consequently it is clear that the part of the fluid contained in the surface OP will be driven along by the part of the fluid contained in the surface JtO, and that the fluid will not remain at rest. Therefore, no part of the body which has been immersed will remain above the surface. How ever, the body will not fall further. For the body has the same heaviness as the fluid and the equivalently placed parts of the fluid compress it similarily. " Proposition IV. — If a body which is lighter than a fluid is placed in this fluid, a part of the body will remain above the surface. (Proof analogous to that of Proposition III,) Proposition V. — If a body which is lighter than a fluid is placed in the fluid, it will be immersed to such an extent that a volume of fluid which is equal to the volume of the part of the body immersed has the same weight as the whole body. The diagram is the same as the preceding one (Proposition III). " Let the liquid be at rest and the body EHTF be lighter than the fluid. If the fluid is at rest, parts which are equivalently placed will be similarly compressed. Then the fluid contained by each of the surfaces XO and OP is compressed by an equal weight. But, if the body BHTC is excluded, the weight of fluid in the first pyramid is equal, with the exclusion of the fluid RSQY, to the weight of fluid in the second pyramid. Therefore it is clear that the weight of the body EHTF is equal to the weight of the fluid RSQ Y. From which it follows that a volume of fluid equal to that of the body which is immersed has the same weight as the whole body. " HELLENIC SCIENCE 31 Proposition VI. — If a body which is lighter than a fluid is totally and forcibly immersed in it, the body will be thrust upwards with a force equal to the difference between its weight and that of an equal volume of fluid. Proposition VII. — If a body is placed in a fluid which is lighter than itself, it wiU fall to the bottom. In the fluid the body will be lighter by an amount which is the weight of the fluid which has the same volume as the body itself. The first Book of the treatise On Floating Bodies concludes with the following hypothesis— " We suppose that bodies which are thrust upwards aU follow the vertical which passes through their centre of gravity. " . . In Book II, Archimedes modified the principle which is the subject of Proposition V, Book I, to the following form— " If any solid magnitude which is lighter than a fluid is immersed in it, the proportion of the weight of the solid to the weight of an equal volume of fluid will be the same as the proportion of the volume of that part of the solid which is submerged to the volume of the whole solid-" - i. i. A i.- A We shall pass over the proof of this proposition, in which Archimedes once more deploys that powerful logical apparatus with which we are now familiar. The rest of Book II is devoted to a detailed study of the equilibrium of a floating body which is shaped like a right segment of a " parabolic conoid. " In Archimedes' language (in the treatise On Conoids and Spheroids), this term refers to the figure which we would now call a parabolic cylinder. It may be surmised that Archi medes was interested in this problem for a most practical reason, for this surface is similar to that of the hull of a ship. It is of interest that, throughout this study, Archimedes approxi mated the free surface of a fluid by a plane, and that he treated verticals an if they were parallel. This is necessary if the concept of centre of gravity is to be utilised. Thus Archimedes must have understood the necessity and the practical importance of this approximation, even though his principle was based on the convergence of the verticals at the centre of the Earth, the spherical symmetry of fluid surfaces and a rather vague hypothesis about the pressures obtaining in the interior of a fluid. CHAPTER TWO ALEXANDRIAN SOURCES AND ARABIC MANUSCRIPTS 1. THE " MECHANICS " OF HERO OF ALEXANDRIA. It seems that Hero of Alexandria lived at some time during the Ilnd Century A. D. His treatise Mechanics discusses certain simple machines — the lever, pulley-block and the screw— alone or in various combinations, and is only available to us in the form of an arabic version which has been translated and published by Carra de Vaux.1 As far as it concerns the history of mechanics, the essential import ance of this work lies in the fact that its author used the now classical idea of moment in his discussion of a lever which was not straight. Whether or not this conception was an original one remains doubtful. Indeed alexandrian learning had access to a treatise of Archimedes that was devoted to levers (TZegJ £vy&v) and in which the problem of the angular lever had been treated. No trace of this writing is extant. However this may be, we shall quote from Hero's Mechanics. " Consider an arm of a balance which does not have the same thickness or heaviness throughout its length. It may be made of any material. It is in equili brium when suspended from the point }' — by equilibrium we understand the arrant of the beam in a stable position, even though it may be inclined in one direction or the other. Now let weights be suspended at some points of the beam— say at b and £. The beam will take up a new position of equilibrium after the weights have been hung on. Archimedes has shown, in this case as 1 Journal asiatique, 1891. ALEXANDRIAN SOURCES AND ARABIC MANUSCRIPTS 33 well, that the relation of the weights to each other is the same as the inverse relation of the respective distances. 1 Concerning these distances in the case of irregular and inclined beams, it should be imagined that a string is allowed to fall from y towards the point £• Construct a line which passes through the point £ — the line 77 £0 — and which should be arranged to intersect the string at right angles. When the beam is at rest the relation of £77 to £0 is the same as the relation of the weight hung at the point e to the weight hung at the point <5. " Hero employed a similar argument in his discussion of the wheel and axle. In fact, in reducing the study of these machines to the principle of circles he was implicitly using the notion of moment. Thus it is clear, though not explicitly stated, that in his discussion of the axle Hero understands that a weight £ can be replaced by an equal force applied tangentially at A, because AF has the same moment as £.2 2. PAPPUS'S THEORIES OF THK INCLINED PLANE AND OF THE CENTRE OF GRAVITY. Pappus (IVth Century A. D.) appears to be the only geometer of Antiquity who took up the problem of the motion arid equilibrium of a heavy body on an inclined plane. The proof that we shall analyse now is taken from Book VIII of his (Collections (From among the varied and delightful problems of mechanics) . Pappus assumes that a certain effort y is necessary to move a weight a on the horizontal plane fiv, and that a power 0 is necessary to draw it 1 OAKKA I>E YAVX'S nurmiHe that Hero in referring to the treatise probably correct. 2 Cf. JorcuET, Lectures de Mvcaniquv, Vol. I, p. 215. Throughout the present book thin treatise will be indicated by the letterH L. M. 34 THE ORIGINS up the inclined plane px. He sets out to determine the relation between y and 0. The weight a on the plane px has the form of a sphere with centre e. Pappus reduces the investigation of the equilibrium of this sphere on Fig. 6 the inclined plane to the following problem. A balance supported at A carries the weight oc at & and the weight ft which is necessary to keep it in equilibrium at y — the end of the horizontal radius eq. The law of the angular lever, which Pappus borrows from Archimedes' UeQl or from Hero's Mechanics, provides the relation On the horizontal plane where the power necessary to move a is y, the power necessary to move along /? will be Pappus then assumes that the power 0 that is able to move the weight oc on the inclined plane px will be the sum of the powers <3 and y, that is ALEXANDRIAN SOURCES AND ARABIC MANUSCRIPTS 35 Evidently the necessity of a power y for pulling the weight a on the horizontal plane derives from Aristotle's dynamics, in which all unnat ural motion requires a motive agency. The argument by which Pappus introduces the lever eArj supporting the two weights a and ft is rather a natural one, even though it does not lead to a correct evaluation of ft. The last hypothesis, concerning the addition of d and y, the powers that are necessary to move ft and a respectively on the horizontal plane is, on the other hand, most strange. However incorrect it may have been, this proof was destined to inspire the geometers of the Renaissance. Guido Ubaldo was to adopt it and Galileo was to be occupied in demonstrating its falsehood. Archimedes certainly formulated a precise definition of centre of gravity, but there is no trace of anything of this kind in those of his writings that are available to us. Therefore it is of some value to record the definition which is due to Pappus. Imagine that a heavy body is suspended by an axis a/? and let it take up its equili brium position. The vertical plane pass ing through oc/J u will cut the body into two parts that are in equilibrium with each other and which will be hung in such a way, on one side of the plane and on the other, as to be equal with respect to their weight. " Taking another axis &.'ftf and repeating the same operation, the second vertical plane n^" passing through a'/?' will certainly cut the first — if it were parallel to it " each of these two planes would divide the body into two equal parts which would be at the same time of equal weight and of unequal weight, which is absurd. " Now suspend the body from a point y and draw the vertical yd through the point of suspension when equilibrium is established. Take a second point of suspension yr and, in the same way, draw the vertical y'd'. The two lines yd, y'd' necessarily intersect. For if not, through each of them a plane could be drawn so as to divide the body into two parts in equilibrium with each other, and in such a way that these two planes were parallel to each other. This is impossible. All lines like yd will therefore intersect at one unique point of the body that is called the centre of gravity. Pappus did not distinguish clearly, as Guido Ubaldo was to do in his Commentary on Archimedes* two books on the equilibrium of weights (1588) between "equiponderant" parts, that is parts that are in equilibrium, in the positions which they occupy, and parts which have the same weight. 36 THE ORIGINS 3. THE FRAGMENTS ATTRIBUTED TO EUCLID IN ARABIC WRITINGS. Greek antiquity does not attribute any work on mechanics to Euclid. However his name occurs frequently in this connection in the writings of arahic authors. Euclid's book on the balance, an arabic manuscript of 970 A. D. which has been brought to light by Dr. Woepke,1 seems to have remain ed unknown to the western Middle Ages. This relic of greek science may be contemporaneous with Euclid and may thus antedate Archimedes. It contains a geometrical proof of the law of levers which is independent of Aristotle's dynamics and which makes explicit appeal to the hypo thesis that the effect of a weight P placed at the end of an arm of a lever is expressed by the product PL. We have had occasion to emphasise the necessity of this hypothesis in our analysis of Archimedes' proof. The treatise Liber Euclidis de gravi et levi, often simply called De ponderoso et Zevi, has been known for a long time. It includes a very precise exposition of aristotelian dynamics which is arranged, after Euclid's style, in the form of definitions and propositions. The latin text renders the terms dvvKjLus and ioyvQ, by which Aristotle meant " power ", as virtus and fortitude. Bodies that travel equal distances in the same medium — air or water — in times which are equal to each other, are said to be equal in virtus. Bodies that travel equal distances in unequal times are of different virtus, and that which takes the shorter time is said to have the greater virtus. Bodies of the same kind are those that, volume for volume, are equal in virtus. That which lias the greater virtus is said to be solidius (more dense). The virtus of bodies of the same kind is proportional to their dimen sions ; that is, the bodies fall with velocities which are proportional to their volume. If two heavy bodies are joined together, the velocity with which the combination will fall will be the sum of the velocities of the separate bodies. Duhem has found, in a XlVth Century manuscript,2 four proposi tions on questions in statics which complete De ponderoso et kvi. This manuscript contains a theory of the roman balance, and shows that the fact that the balance is a heavy homogenous cylinder does not alter the relation of the weights to each other. Finally, in a Xlllth Century manuscript, Duhem has unearthed a text called Liber Euclidis de ponderibus secundum terminorum circonfe- rentiam 3 which connects the law of levers with aristotelian dynamics and also contains a theory of the roman balance. 1 Journal asiatique, Vol. 18, 1851, p. 217. 2 BMiothSque Nationale, Paris, latin collection, Ms. 10,260 3 Ibid., Ms. 16,649. ALEXANDRIAN SOURCES AND ARABIC MANUSCRIPTS 37 4. THE BOOK OF CHARISTION. Liber Charastonis is the latin version of an arabic text due to the geometer Thabit ibn Kurrah (836-901). The original greek version remains unknown, and the question of whether karaston (in Arabic — karstun) refers merely to the roman balance or to the name of the greek geometer Charistion (a contemporary of Philon of Byzantium in the Ilnd Century B. C.) has been the subject of much scholarly debate. We shall follow Duhem l in summarising the theory of the roman balance which is found in Liber Charastonis, <f 9 Fig. 8 A heavy homogeneous cylindrical beam ab whose arms ag and bg are unequal may be maintained in a horizontal position by means of a weight e hung from the end of the shorter arm ag. If bd is the amount by which the longer arm exceeds the shorter arm and u is the centre of &<J, the weight e will be to bd as gu> is to ga. If p is the total weight of the beam db If this weight were known it could be represented exactly by a scale-pan hung from the shorter arm, and the karaston arranged in this way could be treated as a weightless beam. We must also mention, as one of the sources of statics, the treatise De Canoniof a latin translation of a greek text which adds nothing essential to Liber Charastonis. 1 0. S., Vol. T, p. 90. 2 Bibliothvqiie Nationale^ Paris., latin collection, Ms. 737B A. CHAPTER THREE THE XHIth CENTURY THE SCHOOL OF JORDANUS 1. JORDANUS OF NEMORE AND " GRAVITAS SECUNDUM SITUM. " The Middle Ages had access to the Problems of Mechanics and to the works of Aristotle. They had also inherited the fragments attributed to Euclid — with the exception of the Book on the Balance — as well as the Liber Charastonis from arabic learning. They had no knowledge of Archimedes, Hero of Alexandria and Pappus. In spite of the researches of the scholars, the personality of Jordanus remains mysterious. At least three XHIth Century manuscripts on statics have been attributed to him, although these are clearly in the style of different authors. Neither Jordanus's nationality nor the pe riod in which he lived is known with any certainty. Daunou believes him to have lived in Germany about 1050, Chasles associates him with the Xlllth Century while Curtze places him about 1220 under the name of Jordanus Saxo. Michaud has identified him with Raimond Jordan, provost of the church of Uz£s in 1381 which is clearly too late. With Montucla, we shall here adopt the intermediate view that associates Jordanus of Nemore with the Xlllth Century. Like Duhem, we shall follow the Elementa Jordani super demonstra* tionem ponderis.1 This work comprised seven axioms or definitions followed by nine propositions. The essential originality of Jordanus lay in the systematic use, in his study of the motion of heavy bodies, of the effective path in a vertical direction as a measure of the effect of a weight, which was usually placed at the end of a lever and described a circle m consequence. Thus his statics stems, implicitly, from the principle of virtual work. The word work, taken in the modern sense, is to be con- 1 Bibliothdque Nationals, Paris, Ms. 10,252, dated 1464. There also exists an in complete manuscript of the same work, dating from the Xlllth Century, in the Biblio- theque Mazarine, Ms. 3642. THE XlHth CENTURY 39 trasted with the word velocity and with the concept of virtual velocities which may be traced in the arguments of Problems of Mechanics. Of course Jordanus never used the word " work " itself. He considered the heaviness of a particle relative to its situation (gravitas secundum situm) without making clear the relation that exists between this quan tity and the heaviness in the strict sense. Jordanus formulated his principle in a picturesque Latin which merits quotation. " Omnis ponderosi motum esse ad medium, virtutemque ipsius poten- tiam ad inferiora tendendi et motui contrario resistendi. " Gravius esse in descendendo quando ejusdem motus ad medium rectior. " Secundum situm gravius, quando in eodem situ minus obliquus est descensus. 64 Obliquiorem autem descensum in eadem quantitate minus capere de directo. " Or— " The motion of all heavy things is towards the centre,1 its strength being the power which it has of tending downwards and of resisting a contrary motion. " A moving body is the heavier in its descent as its motion towards the centre is the more direct. 46 A body is the heavier because of its situation as, in that situation, its descent is the less oblique. 44 A more oblique descent is one that, for the same path, takes less of the direct. " Thus a certain weight placed at 6, at the end of the lever c6, has a smaller gravity secundum situm than the same weight has when it is at a, at the end of the horizontal radius ca. Indeed, on the circumference of the circle with centre c and radius ca =-- c6, if the body falls from b to h along the arc oh the effective path in a vertical direction is b' hf. On, the other hand if the body starts from a and falls along an^arc c5, which is equal to the arc bh> the effective vertical path is czr and is greater than V h*. Thus the descent 6A, equal to the descent oz, is more oblique than that and takes less of the direct. 1 Understood as the common centre of all heavy things in Aristotle's sense. 40 THE ORIGINS This idea led Jordamis to a proof of the rule of the equilibrium of the straight lever whose originality cannot be contested. h b Fig, 10 46 Let acb be the beam, a and b the weights that it carries, and suppose that the relation of b to a is the same as that of ca to c6. I maintain that this rule will not change its place. Indeed, if the arm supporting b falls and the beam takes up the position c?ce, the weight 6 will descend by he and a will rise by fd. If a weight equal to the weight b is placed at I, at a distance such that d = c6, this will rise in the motion by gm = he. But it is clear that dfis to mg as the weight Z is to the weight a. Consequently, what is sufficient to bring a to A will be sufficient to bring 1 to m. But we have shown that b and I counterbalance each other exact ly, so that the supposed motion is impossible. This will also be true of the inverse motion. " Duhem writes in this connection I — " Underlying this demonstration of Jordanus the following principle is clearly evident— that which can lift a weight to a certain height can also lift a weight which is k times as great to a height which is k times less. This principle is then the same as that which Descartes took as a basis for his complete theory of statics and which, thanks to John Bernoulli, became the principle of virtual work. " Jordanus was less fortunate when he turned his attention to the angular lever. He considered a lever acf carrying equal weights at a and /which were placed in such positions that ac = ef. JordanuH was of the opinion that, under these Fig. 11 1 o. s., Vol. r, P. 121. THE Xlllth CENTURY 41 conditions, a would dominate /. He arrived at this conclusion by£con- sidering two equal arcs al and mf. It is clear that the " direct " taken by the weight a is greater than the " direct " taken by the weight /. This incorrect conclusion is obtained because, since the linkages are rigid, the two displacements al andjfm are not simultaneously possible. Jordanus thus misunderstood the idea of moment. As early as the Xlllth Century the Elementa Jordani were generally united by the copyists with De Canonio to form the Liber Euclidis de ponderibus.* This artificial associations and this imaginative titles are the despair of the scholars and it has needed all the learning of Duhem to elucidate them. Every truly novel idea evokes a reaction. The Elementa Jordani did not provide an exception to this rule. In the Xlllth Century a critic wrote a commentary of Jordanus which Duhem calls the Peripa tetic Commentary.* This author at every turn invokes the authority of Aristotle and has scruples about applying the gravitas secundum situm to a motionless point — in modern language, about making appeal to a virtual displacement. It does not appease his conscience to consider that rest is the end of motion. " The scientific value of the Commen tary is nothing, " declares Duhem.3 " But its influence did not disappear for a very long time, and even the great geometers Tarta- glia, Guido Ubaldo and Mersenne had not entirely freed themselves from it. " 2. TlIE ANONYMOUS AUTHOR OF " LlBER JORDANI DK RATIONE PON- DERIS. " TlIE ANGULAR LEVKR. TlIE INCLINED PLANE. We now come to an especially noteworthy work which figures in the same manuscript as the Peripatetic Commentary under the title Liber Jordani de ratione pondrris, and which did not remain unknown in the Renaissance, Tartaglia sent it to CurtiuH Trojanus who published it in 1565. This work supereedes and rectifies the Klementa Jordani on many important points. All the same, it is based on the same principle of gnivitas secundum situm. Duhem, who brought this manuscript to light, terms the anonymous author a u Precursor of Leonardo da Vinci. " Indeed, in many respects this precursor surpassed Leonardo, who, for example, spent himself in fruitless efforts to evaluate the apparent weight of a body on an inclined plane. It seems more natural to simply speak of an anonymous Nationals, Paris, latin collection, MHH. 7310 and 10,260, 2 Ibid., Ms. 7378 A. 3 0. S., Vol. I, p. 13k 42 THE ORIGINS author of the Xlllth Century, a disciple of Jordanus who had out stripped his master. In connection with the bent lever this author corrected Jordanus's error. As before, let a lever acf carry equal weights at a and f and be placed in such a position that aaf — ff'. Fig. 12 It is impossible that the weight a should dominate the weight /. For if two arcs aft, j#, are considered on the two circles drawn through a and / and corresponding to equal angles ach and jfr£ the descent of a along rh necessitates that the equal weight at /should rise through a distance In which is greater than rh. This is impossible. In the same way it can be seen that / will not dominate a. For if the arcs/E and am correspond to equal angles fcx and acm, the descent of /along tx makes it necessary that the equal weight placed at a should rise by pm, which is greater than tx. This is impos sible. Therefore there is equilibrium in the position considered, in which aa'^ff'. The anonymous author generalised this result to an angular balance carrying unequal weights at a and 6, Fig. 13 THE XHIth CENTURY 43 and obtained the result that in equilibrium it is necessary that the distances aaf and 66' from a and 6 to the vertical drawn through the point of support, c, are in inverse ratio to the weights a and 6. We see that this author knew and used the notion of moment. Elsewhere he wrote on this subject, " If a load is lifted and the length of its support is known, it can be determined how much this load weighs in all positions. The weight of the load carried at e by the support be will be to the weight carried at / by fb as el is to fr or as pb is to xb. A weight placed at e, at the end of the lever 6e, will weigh as if it were at u on the lever 6/. " Thus the idea of gravitas secundum situm, which Jordanus had used qua litatively, became precise. Our anonymous author also con cerned himself with the stability of the balance, and rectified certain errors which were contained in the relevant parts of Problems of Mecha nics. x p o More than this, he resolved the problem of the equilibrium of a heavy l^* body on an inclined plane, a problem which had eluded the wisdom of the greek and alexandrian geometers. In order that this may be done, it is first observed that the gravitas secundum situm of a weight on an inclined plane is independent of its position on the plane. The author then attempts a comparison of the value that that gravity takes on differently inclined planes. We shall quote from Duhetn's translation of this same Xlllth Century manuscript. u If two weights descend by differently inclined paths, and if they are directly proportional to the declinations, they will be of the same strength in their descent. " Let ab be a horizontal and W, a vertical. Suppose that two oblique lines da and dc fall on one side and on the other of 6rf, and that dc IISLS the greater relative obliquity. By the relation of the obli quities I understand the relation of the declinations and not the relation of the angles ; this means the relation of the lengths of the named lines counted as far as their intersection with the horizontal, in such a way that they take simUarily of the direct. ** In the second place, let e and h be the weights placed on dc and da respectively, and suppose that the weight e is to the weight h as 44 THE ORIGINS dc is to da. I maintain that in such a situation the two weights will have the same strength. " Indeed, let dk be a line having the same obliquity as dc and, on that line, let there be a weight g which is equal to e. " Suppose that the weight e should descend to Z, if that is possible, and that it should draw the weight h to m. (It is clear that the author imagines the two weights to be connected by a thread which passes over a pulley at d.) Take gn equal to Am, and consequently equal to el. Draw a perpendicular to db which passes through g and A, say ghy. Drop a perpendicular It from the point I onto db. Then drop [the perpendiculars] nr, mx, and ez. The relation of nr to ng is that of dy to dg and also that of db to dk. Therefore mx is to nr as dk is to da ; that is to say, as the weight g is to the weight h. But as c is not able to pull g up to n (nr = ez), it is no better able to pull h up to m. The weights therefore remain in equilibrium. " This demonstration, which leads to the correct law of the apparent heaviness of a body on an inclined plane, was directly inspired by that of Jordanus concerning the equilibrium of a straight lever* Like that, it implicitly proceeds according to the principle of virtual work. We shall now give some indication of the ideas on dynamics which were used by the author of Liber Jordani de ratione pondcris. The environment's resistance to the motion of a body depends on the shape of the body, which penetrates the environment the better as its shape is the more acute and its figure the more smooth. It depends, in the second place, on the density of the fluid traversed. AH media are compressible ; the lower strata, compressed by the upper ones, are the denser and those which hinder motions more. At the front of the moving body will be a part of the medium compressed on, and sticking to it. But the other parts of the medium, which are displaced by the moving body, curl round behind to occupy the THE XHIth CENTURY 45 space which the body has left empty. This motion of lateral parts of the medium may be compared to the bending of an arc. The more heavy the medium is at traversal, the slower is the des cent of a heavy body. The descent is slower in a fluid which is more dense. Greater width diminishes the gravity. A heavy thing will move more freely as the duration of its fall in creases. " This is more true in air than in water, because air is suited to all kinds of motion. Thus a falling body drags with it, from the outset of its motion, the fluid that lies behind it and sets in motion the fluid in its immediate contact. The parts of the medium set in motion in this way, in their turn move those that adjoin them, in such a way that the latter, which are already in motion, present a lesser obstacle to the falling body. For this reason the body becomes heavier and imparts a greater impulsion to the parts of the medium which it dis places until these are no longer simply pushed by the body, but drag it along with them. Thus it happens that the gravity of a moving body is increased by their traction and that, reciprocally, their motion is multiplied by this gravity so that it continually increases the velocity of the body. " The shape of a heavy body affects the strength of its weight. The strength of a motive agency seems to be equally baulked by a body's very large or very small weight. Rotation of a propellant increases its strength, and does so more effectively as the radius is greater. A body whose parts are coherent is thrown directly backwards if it is stopped by a collision during its motion. " The parts of a moving body A that He in front are the first to meet the obstacle C. They are therefore compressed by the mass and the impetuosity of the parts which lie behind them, and are forced to condense. The impetuosity of the parts behind is annulled in this way. The parts in front now assume their original volume again and recoil backwards, thus com municating an impulsion to the others. If the parts which are com pressed in this way are able to detach themselves from each other they will be thrown off in one direction and another. ** If the heaviness of a body is not uniform, the denser part will place itself in front, whatever the part to which the impulsion is given.1 These ideas on dynamics held by Jordanus's School are much less interesting and moreover, less original than its statics. We have cited them here as curiosities. 1 Cf., DUHEM, £tude$ sur Leonard de Vinci^ Series I (Hermann), 1906* 46 THE ORIGINS From the historical point of view it must be remarked that Duhem, in writing his first studies on the origin of statics, first believed the work of this unknown disciple of Jordanus to be entirely original. But the later discovery of a XHIth Century manuscript1 led him to a treatise De Ponderibus which was more complete than the Liber Jordani de ratione ponderis. Now this treatise, divided into four books, seems to be a complex in which various works have been artificially united. There is first a book, of indisputable Medieval origin, that repeats the demonstra tions Jordanus used and supplements them by the condition for the equilibrium of the angular lever and the determination of the apparent weight of a body on an inclined plane. The second book appears to have been inspired by De Canonio while the third treats the concept of moment and the conditions for the stability of the balance. Finally there is a fourth book devoted to dynamics. The last two books are closely related to Problems of Mechanics although they alter, correct and complete this work in many places. Certain indications led Duhem to surmise that the two books are a relic of greek science and were probably handed on by the Arabs — this because no latinised greek terms are found in them.2 Accord ingly it is possible that our unknown author did not discover the idea of moment himself. This limits the originality of his work, but it remains that gravitas secundum situm properly belongs to the Xlllth Century School, and that it was used by this School to obtain a correct solution of the problem of the inclined plane long before Stevin and Galileo did so, 1 Biblioth&que Nationale, latin collection, Ms. 8680 A. 2 C/. DUHEM, 0. S., Vol. II, note F, p. 318. LEONARDO DA VINCI himself seema to have been unaware of the three last books of De Poncferi&us— another argument for not regarding the unknown author as his precursor. CHAPTER FOUR THE XlVth CENTURY THE SCHOOLS OF BURIDAN AND ALBERT OF SAXONY NICOLE ORESME AND THE OXFORD SCHOOL 1. THE DOCTRINE OF " IMPETUS " (JOHN The idea of attributing a certain energy to a moving body solely on account of its motion is entirely foreign to aristotelian dynamics. In Antiquity John of Alexandria — surnamed Philopon — who lived in the Vth Century A. C., was alone in disputing Aristotle's belief in this matter. Thus he held that the air which was set in motion could not become the motive agency of a projectile, whose motion was, on the contrary, easier in a vacuum than in air. " Whoever throws a projectile embodies in it a certain action, a certain power of self-move ment which is incorporated. . . . Nothing prevents a man from throwing a stone or an arrow even when there is no other medium than the vacuum. A medium hinders the motion of projectiles, which cannot advance without dividing it — nevertheless they can move through these media* Nothing therefore prevents an arrow, a stone or any other body from being thrown in the vacuum. Indeed, the motive agency, the moving body and the space that will receive the projectile are all present."1 Philopon's thesis was handed on to the Middle Ages by the Arabs — in particular, by the astronomer Al Bitrogi, But while assuming the existence of a " property which remains attached to a stone or an arrow after the projectile has been thrown, " he held that this property decreased at such a rate and to such an extent as the projectile was separated from its motive agency. Albertus Magnus and Saint Thomas Aquinas knew of this tradition but did not give the least credit to John Philopon's argument. For example, Saint Thomas Aquinas believed that if the existence of an 1 Erudissima commentaria in prim&s quatuor Aristotelis de naturali auscultations libros* Venice (1532), Trans. DUHEM, 48 THE ORIGINS apparent property impressed on a moving body were assumed, " violent motion would arise from an intrinsic property of a moving body, which is contrary to the very notion of violent motion. Moreover, it would follow from this that a stone would be altered in its substantial form by the very fact that it moved from place to place, which is contrary to common sense. " I Roger Bacon, Walter Burley and John of Jandun all adopted Aristotle's doctrine on this matter. The first Schoolman to oppose this opinion was William of Ockham (1300-1350). He asked himself where the motive agency might be. It cannot reside in the apparatus or organism that has thrown the projectile, for this apparatus can be destroyed immediately after the launching without interupting the pro gress of the projectile. Nor can the motive agency be the air which is set in motion. For the arrows of two archers which are shot towards each other can be arranged to collide with each other, which requires that the same air produces two different motions at the same time. There cannot be distinguished elsewhere a cause that could provide the motive power. Such a cause cannot reside in the launching appa ratus nor in the motion of the projectile itself. If something which is its own motive agency is thrown, that which moves the body cannot be distinguished from the moving body itself. Moreover, motion from place to place is not something which is renewed at each instant, requir ing the constant presence of a motive cause. It is true that the pro jectile passes through a different region at each instant, but this does not in itself constitute anything novel. It is only novel with respect to the moving body.2 Thus William of Ockham decided to reject Aristotle's axiom which requires the continuous existence of a motive agency in contact with, yet not part of, the projectile. He did not, however, replace it by any new principle. We now arrive at the doctrine of impetus that was conceived by Buridan. John I. Buridan, of Bethune, was rector of the University of Paris in 1327 and canon of Arras in 1342. He died in Paris after 1358.:i In a memoir called Quaestiones octavi libri physicorum^ Buridan 1 Opera omnia, Vol. Ill — Commentaria in libros Aristotclis de Caclo ft Mundo* Book III, lect. VII. 2 Cf. DUBDEM, Etudes sur Leonard de Vinci, Series II, p. 192. 3 DUHEM, who has studied BURIDAN'S works in detail, including those concerning free will, says that he has found no trace of the parable of the ass, which apart from IUH status in the history of mechanics, has made Buridan's name classical. 4 BiUiotheque Nationale, Paris, latin collection, Ms. 14,723, fol 106-107. In the text we are following DUHEM'S translation. THE XlVth CENTURY 49 discussed the scholastic thesis of the motion of projectiles. Aristotle, he says, mentions two opinions on this matter. The first invokes &vTineQiaTcx.ai,<;. As a projectile moves rapidly away from its position, Nature, who does not allow the existence of a vacuum, makes the air behind the projectile rush in towards this position with the same velocity. This air pushes the projectile and the same effect is reproduced, at least for a certain distance. This opinion is rejected by Aristotle — if no other principle than &.VT metier &oi<; is invoked, it is necessary that all bodies which are behind the particle, including the sky itself, participate in the projectile's motion. Indeed, the air will also leave its position. It is then necessary that another body must replace it, and so on in an indefinite sequence, unless it is assumed that a certain rarefaction of bodies behind the projectile is produced. According to the second opinion, which Aristotle seems to have supported, the launching of a projectile disturbs the ambient air at the same time. This air, violently set in motion, has, in its turn, the power to move the projectile. The first mass of air moves the projectile until this comes to a second mass of air. This second one moves it until a third is reached, and so on. Further, Aristotle is heard to say, there is not merely a single moving body, but sxiccessive moving bodies, a series of consecutive or contiguous motions. Buridan set the following observations against these theories. A top or a grindstone will turn for a very long time without leaving its position, in such a way that the air does not have to follow it to fill an abandoned place. Further, the wheel will continue to txirn if it is covered and thus separated from the surrounding air. A javelin whose following end is armed with a point as sharp as its tip will move as rapidly as if this were not tapered at the back. Now since the air is easily divided by the javelin's sharpness, it cannot push strongly on this backward pointed part. A ship will continue to move for a long time after towing has been stopped, and a boatman will not feel the air pushing it — on the contrary, he feels the air slowing down the ship's motion. The air set in motion should be able to move a feather more easily than a stone. Now we are not able to throw a feather as far as a stone. Buridan himself put forward the following thesis. " Whenever some agency sets a body in motion, it imparts to it a certain impetus, a certain power which is able to move the body along in the direction imposed upon it at the outset, whether thia be upwards, downwards, to the side or in a circle. The greater the velocity that the body is given by the motive agency, the more powerful will be the impetus which is given to it. It is this impetus which moves a stone 50 THE ORIGINS after it lias been thrown until the motion is at an end. But because of the resistance of the air and also because of the heaviness, which inclines the motion of the stone in a direction different from that in which the impetus is effective, this impetus continually decreases. Con sequently the motion of the stone slows down without interruption. Finally the impetus is overcome and destroyed at the point where gravity dominates it, and henceforth the latter moves the stone towards its natural place. . . . " All natural forms and dispositions are received by matter in pro portion to itself. Consequently the more matter a body contains, the more impetus can be imparted to it, and the greater is the intensity with which it can receive the impetus. ... A feather receives such a weak impetus that this is immediately destroyed by the resistance of the air. In the same way, if someone throws projectiles and sets in motion with equal velocities a piece of wood and a piece of iron, which have the same volume and the same shape, the piece of iron will travel further because the impetus which is imparted to it is stronger. It is for the same reason that it is more difficult to stop a large blacksmith's wheel, moving rapidly, than a smaller one. ..." In short the impetus, in Buridan's sense, increases with the velocity. In addition, it is proportional to the density and to the volume of the body concerned. Further, in Buraidan's view, the existence of impetus explained the acceleration of falling bodies. " The existence of impetus seems to be the cause by which the natural fall of bodies accelerates indefinitely. At the beginning of the fall, indeed, the body is moved by gravity alone. Therefore it falls more slowly. But before long this gravity imparts a certain impetus to the heavy body — an impetus which is effective in moving the body at the same time as gravity does. Therefore the motion becomes more rapid. But the more rapid it becomes, the more intense the impetus becomes. Therefore it can be seen that the motion will be accelerated con tinuously. " Further, Buridan applied the notion of impetus to stars as well as to terrestrial bodies. 46 In the Bible there is no evidence of the existence of intelligences charged with communicating their appropriate motion to the heavenly bodies. It is therefore permissible to show that there is no necessity to suppose the existence of such intelligences. Indeed it can be said that when He created the World, God set each of the heavenly bodies in motion in the way that he had chosen — that He imparted to each of them an impetus which has kept it moving ever since. Thus God no longer has to move these bodies, except for a general influence similar THE XlVth CENTURY 51 to that by "which He gives his consent to all things that occur. It is for this reason that, on the seventh day, He was able to rest from the tasks which He had accomplished and to confine himself to the creation of things concerning mutual actions and feelings. The impetus that God imparted to the heavenly bodies is neither weakened nor destroyed by the passage of time. For in these heavenly bodies there are no ten dencies towards other motions and because, moreover, there is no longer any resistance which could corrupt and repress this impetus. I do not say all this with complete assurance. I would only ask the theologians to show me how all these things happen. " As a true Scholastic Buridan believed himself obliged to defend the doctrine of impetus from the metaphysical objections that could be advanced against it. The motion of a projectile is a violent one in Aristotle's sense. Now, according to the Ethics (Book III), violent phenomena stem from an extrinsic, not an intrinsic, cause. To this Buridan replied that the impetus of a moving body is effectively violent, not natural. The nature of heavy things favours a different motion and the destruction of the impetus. On the question of whether the impetus is distinct from the motion and whether it is of a permanent kind, Buridan replied that impetus could not itself be motion because all motion requires a motive agency ; that impetus was a permanent reality, distinct from the local motion of the projectile ; and that it was probable that the impetus was a quality whose nature was to actuate the body to which it was imparted. These subtleties add nothing to Buridan's positive doctrine. It is more important to remark that Buridan maintained that the impetus lasted indefinitely if it was not diminished by a resistance of the medium or modified by some agency affecting the moving body. This is the germ of the modern principle of inertia. 2. TlIE SPHERICITY OF THE EARTH AND THE OCEANS— ALBERT OF SAXONY AND THE ARLSTOTLETIAN TRADITION. H In the first chapter of this book we referred to the a priori, or physical proofs, and the a posteriori proofs which Aristotle gave of the sphericity of the Earth and the oceans. For better or worse, tradition preserved and enriched these proofs. Pliny the Elder, in his Natural History, supplemented Aristotle's evidence with facts that strictly derive from capillarity — the sphericity of drops of water, the meniscuses of liquids, etc. . * . Ptolemy only retained the a posteriori proofs which Aristotle had given. Simplicius, in his commentary on De CaeJo, corrected the dimensions attributed to 52 THE ORIGINS the Earth after Erastosthenes' evaluation. Averroes confined himself to an elaboration of Aristotle's evidence. John Sacro Bosco — the author of a treatise called De Sphaera which became the most widely known cosmography in the Xlllth Century — reproduced Ptolemy's account. Albertus Magnus firmly excluded the evidence depending on the sphericity of water drops. Saint Thomas Aquinas limited himself to Aristotle's proofs alone, while Roger Bacon supplemented them with the following corollary which was acclaimed by the Schoolmen — any given vessel will contain a smaller quantity of liquid as it is taken further from the centre of the Earth. We now come to Albert of Rickmersdorf, called Albert of Saxony. Though his biography is somewhat mysterious, it is certain that he was enlisted at the Sorbonne from 1350 to 1361 and that he was rector of the University of Paris from 1353. His Acutissimae Quaestiones on Aristotle's Physics had considerable repercussions, and his influence was felt by most students of mechanics, including Galileo himself. Albert of Saxony suggested going back to the measurement of a degree of meridian at different latitudes in order to determine the true figure of the Earth. (This idea was applied by John Femel at the beginning of the XVIth Century and, of necessity, repeated in the XVIIth Century.) " If these two paths are found to be equal this is a certain indication that the Earth is circular from north to south. If on the contrary, it were found that they lacked equality this would be an indication that the Earth was not round from north to south. " Like Albertus Magnus, Albert of Saxony excluded the evidence pro vided by small drops, which is common to all liquids, like mercury, and is especially noticeable in small quantities. Albert of Saxony stated the following corollaries, which were to become popular among the Schoolmen. " 1. From the fact that the Earth is round it follows that lines normal to the surface of the Earth will approach each other continuously, and meet at the centre. "2. It follows that if two vertical towers are built, the higher they become, the further away from each other they will be ; and that the deeper they are, the nearer together they will be. " 3. If a well is dug with a plumb-line, it will be larger near the opening than at the bottom. " 4. Any line such that all its points are at an equal distance front the centre is a curved line. If a straight line touches the Earth's surface at its middle point, this point will be nearer to the centre than the ends of the line. It follows that if a man goes along this straight line, he THE XFVth CENTURY 53 descend for a time and then will rise. He will descend, indeed, until lie has come to the point which is nearest to the centre of the Earth and will rise from the moment that he leaves that point behind him. " From this it can be concluded that a body which describes a tra jectory between two fixed ends, a trajectory which either rises or falls without interruption, must necessarily travel a shorter distance than if the path went from one point to the other without rising or falling. This is seen clearly if it is supposed that the first trajectory is a diameter of the Earth and the second is a half- circumference having this diameter as chord. " 5. When a man walks on the surface of the Earth his head moves more quickly than his feet. . . . One can conceive of a man so tall that his head moves in the air twice as quickly as his feet move over the ground. " These paradoxes are typical of the scholastic attitude of mind and it is for this reason that we have quoted them. Undoubtedly they were intended to stimulate the minds of students, and perhaps, too, to confuse those who were not scholars. The dialectic of the Schoolmen was not in the least concerned with orders of magnitude. It was amusing to proliferate the consequences of the convergence of verticals and their practical parallelism was of no concern — that was a notion suitable for craftsmen. And these, in their turn, were not much worried by the comments of the Schoolmen when they were building their towers and digging their wells according to the simple rules of their practice. 3, ALBERT OF SAXONY'S THEORY OF CENTRE OF GRAVITY. When commenting on that thesis of Aristotle according to which there exists, in each heavy body, a centre of gravity (to IJL&GQV) which tends to be carried towards the centre of the Universe, Albert of Saxony specified that " each of the parts of a heavy body is not moved in such a way that its own centre would come to the centre of the World, for this would be impossible. Rather it is the whole body which falls in such a way that its centre would become the centre of the World. It is false, and contrary to observation, to say that a large body falls more slowly than a lighter body, or that ten stones which are united together hinder each other's fall. " The Earth, limited partly by the concave surface of the water and partly by the concave surface of the air, is in ita natural position when its centre of gravity is at the centre of the World. If this is not so, it 54 THE ORIGINS will start falling and will move until the centre of the aggregate which it forms with all the other heavy bodies becomes the centre of the World. It should be remarked, as Jouguet has done in this connection,1 that Albert of Saxony's concept of centre of gravity, the point of a body at which all the weight appears to be concentrated, was a purely experi mental one, to Mm and his School. It was not the same as the modern conception of centre of gravity, which depends on the approximation that verticals are parallel. On the contrary, it was developed together with a systematic consideration of the convergence of verticals which was carried to the point of paradox, as we have seen. This co-existence was at the root of several fallacies which were to perplex people, even such eminent ones as Fermat, until the XVIIth Century.2 We return to Albert of Saxony. Suppose that the Earth is displaced from its natural place — for example, to the concavity of the orbit of the Moon — and held there by force. Suppose that, there, a heavy body is allowed to fall. Then this body will be attracted towards the centre of the World, not towards the centre of the Earth. " If heavy bodies move towards the ground, this is in no way caused by the Earth, but happens because they approach the centre of the World by going to wards the Earth. " The Earth does not have a uniform gravity — " the part which is not covered by sea, being exposed to the rays of the Sun, is more dilated than the part the waters cover. Besides, if its geometrical centre were to coincide with its centre of gravity, and consequently with the centre of the World, it would be entirely covered by the waters. " Here, in Albert of Saxony's writings, is the trace of an argument that had preoccupied some of his immediate predecessors. If all elements, declared Walter Burley (1275-1357), had the form of spheres with centres at the centre of the Universe, each would be in its natural place — but then the Earth would be completely covered with water. John Duns Scot (1275-1308) resolved this difficulty, in his Doctor Subtilis, with a finalist explanation — to witt, a part of the Earth is uncovered with a view to the safety of living beings. Albert of Saxony believed therefore that it was the Earth's centre, of gravity, not its geometrical centre, that was placed at the centre of the World. Furthermore, the Earth was not fixed in position. A host of reasons, such as heating by rays of the Sun, could produce a variation of the distribution of gravity in the terrestrial mass, and could 1 JOUGUET, L. M., Vol. I, p. 60. 2 This question was at the root of the controversy on Geostatics, to which we shall return. THE XlVth CENTURY 55 displace its centre of gravity. As a more substantial mechanism, Albert of Saxony mentioned erosion. The question arose as to how the mass of the waters could be intro duced. On this point Albert of Saxony's opinion was somewhat variable. In commenting on the Physics he wrote, " What I have written about the Earth alone may be understood equally for the whole aggre gate formed by the Earth and the waters. These two elements undoub tedly form a total and unique gravity whose centre of gravity is at the centre of the World. " At this same centre of the World was also to be found the centre of lightness of light bodies. It is this that explains the following picture which he boldly painted. " Since the cold is especially intense at the poles, the layer of igneous element there must be thinner than at the equator if fire, which is con tinuously created at the equator is not to run towards the poles. In the same way that water constantly runs towards lower places in order that the centre of all gravity shall be at the centre of the World, so we must assume that fire travels, without interruption, from the equator towards the poles in order that its centre of lightness shall be at the centre of the World. " It should be imagined that, at the poles, fire is constantly being transformed into air, and at the equator air is constantly being trans formed into fire ; and that fire continually runs from the equator to wards the poles in order that the centre of all lightness, like the centre of gravity, shall be found at the centre of the World. " In short, as Duhem has observed,1 "the common centre of heavy bodies — both the closed earth and the water — and the common centre of light bodies — both air and fire — are placed at the centre of the World. " However, in commenting on De Caelo, Albert of Saxony took a different view. " We reply by denying that the centre of the World coincides with the centre of the aggregate formed by the earth and the water. Indeed, if it is imagined that all the water were lifted off, the centre of gravity of the Earth would still be at the centre of the World. . . . For, essen tially, the earth is heavier than water. Therefore, whatever may be the quantity of water which is found on one side of the Earth and not on the other, this part of the Earth will in no way receive more help than previously in counterpoising and pushing awy the other part. ..." It is explained " that one part of the Earth rises out of the waters. The Earth, indeed, is not uniformly heavy, so that its centre of gravity 1 DUHEM, 0. $., Vol. II, p. 28. 56 THE ORIGINS is placed at a great distance from its geometrical centre. The centre of gravity is much nearer one of the convex hemispheres that define the Earth than the other. Therefore the water, which is unifirmly dense and tends towards the centre of the World, runs towards that part of the terrestrial sphere that is nearest the centre of gravity of the Earth, so that the other hemisphere, that which is further distant from the centre of gravity, remains uncovered. " The weakness of this argument is clear. But undoubtedly the theory was in harmony with the belief, common at that time, in the existence of a terrestrial hemisphere completely covered by a vast ocean. It is paradoxical to see Albert of Saxony thus holding that the waters of the sea do not exert any heaviness, but this is in accord with a more general thesis that is indicated below. Albert of Saxony distinguished between heaviness in the potential state, that of a heavy body occupying its natural place, and the actual heaviness that sets a body in motion when it has been displaced from its natural place (or shows itself as a resistance to obstacles which oppose the body's motion). We shall quote from Duhem's commentary. " The parts of a heavy body, be they solid or liquid, do not push the adjacent parts when they are in their natural place, since their heaviness remains in its potential state. Thus the bottom of the sea does not support any load or any pressure that is due to the water above it. In all circumstances the strength of the heaviness, whether it is habitual or actual*, has the same magnitude in the same heavy body. A part of the earth inclines towards its natural place just as much if it is placed higher up than if it is lower down. " It is clear that this thesis contradicts the fundamental axiom of Jordanus — Gravius esse in descendendo quando ejusdem motus ad medium rectior. Moreover, it is not surprising that Albert of Saxony should have rejected the idea of gravitas secundum situm, and have substituted for it the concept of a greater or smaller resistance of the supporting medium to the fall of a moving body. 4. ALBERT OF SAXONY'S KINEMATICS. THE ACCELERATION OF FALUN*; BODIES. Whether explicitly or not, the physicists and astronomers of Anti quity treated only the simple uniform motions of translation and rota tion, and confined themselves to a simple qualitative description of accelerated motion. THE XlVth CENTURY 57 In a Xlllth Century manuscript1 there is the statement that it is correct to ascribe the velocity of its mid-point to a radius turning about its centre. This text is mentioned by Thomas Bradwardine, proctor of the University of Oxford, in his Tractatus Proportionum (1328). Bradwardine denies this statement and attributes the velocity of its most rapidly moving point to a body in uniform rotation. Albert of Saxony stated these two opinions and supported that of Bradwardine. To set against this, he gave a correct definition of the angular velocity of rotation (velocitas circuitionis) . Further, he distin guished between deformed motions, in which the velocity of a moving body varies from one point to another, and irregular motions, in which the velocity varies from one instant to another. In Book II, paragraph XIII of his Quaestiones? Albert of Saxony examined two possible laws which might govern the fall of bodies — an increase of velocity which is proportional to the distance travelled and an increase proportional to the time taken. In another place, he rejects these two laws, which lead to velocities which become infinite with the distance travelled or the time taken, and contemplates a law which would necessitate that the velocity approach a finite limit when the time increases indefinitely. On this occasion, Albert of Saxony declares himself a supporter of the doctrine of impetus in order to explain the acceleration of falling bodies. He observes, however, that the resistance, increasing more quickly than the impetus is acquired, will limit the velocity of the moving body. It is important to notice this connection between Albert of Saxony and John Buridan's doctrine, and to recognise the considerable authority which the latter's work had over this long period. 5. THE DISCUSSION OF ACTION AT A DISTANCE. In commenting on Aristotle, AverroSs and Albertus Magnus had held that the weight of a heavy body did not vary with its distance from the centre of the World. On the other hand Saint Thomas Aquinas, arguing from the acceleration of falling bodies, assumed that a heavy body increased in weight as it approached this centre. In the XlVth Century John of Jandun, in his commentary on De Caelo (Book IV, para. XIX), declared himself for the first opinion. The natural place cannot be the " motor " of a heavy body, because the motor must always accompany the moving body. It is not possible to have action at a distance. The attraction of iron by a magnet presumes 1 Biblioth&que National^ Paris, latin collection, Ms. 8680 A. 58 THE ORIGINS the alteration of the medium, the propagation of a species magnetica. On the other hand, William of Oekham denied that the motive agency should always accompany the moving body. He declared that iron is attracted at a distance by a magnet without the intermediary of any quality, either in the medium or the iron. He assumed that the magnet was, in itself, the total cause of the effect. In his Quaestiones (Book III, para. VII), Albert of Saxony held that the effect of a body's natural place on a body was different from the action of a magnet on iron. It is true that a heavy body accelerates in falling,, " but its initial1 velocity is not greater when it is close to its natural place than when it is separated from it. " Thus the gravity doeks not depend on the distance from the centre of the World — the attraction of a magnet, on the other hand, vanishes at some distance. The Schoolmen of the XlVth Century therefore rejected the hypo thesis that ascribed weight to an attraction at a distance exerted by the centre of the Earth. But, as Duhem has observed, " in order to prove this hypothesis wrong, it was necessary to work out its conse quences. They [the Schoolmen] discovered that, on the basis of this supposition, the weight of a body would vary with its distance from the centre of attraction. From this, it was argued that the body would have, in falling, an initial velocity1 which was less if its starting-point were further away from the centre. " 2 These discussions on the attraction and even the more metaphysical argument about the plurality of worlds made the copernican revolution, to a certain extent, possible. 6. NICOLE ORESME — A DISCIPLE OF BURIDAN. From 1348, Nicole Oresme, of the diocese of Bayeux, was a student of theology. In 1362 he was grand master of the College of Navarre. He became Bishop of Lisieux on August 3rd, 1377 and died there on July llth, 1382. Charles V entrusted Oresme with the task of translating (into French) and annotating certain of Aristotle's works which had previously only been accessible to the scholars. The four books of On the Heavens and the World were included in the commission, though this particular part 1 The question here is one of a free fall — the initial velocity in the modern Berne in therefore zero. The velocity which ALBERT OF SAXONY intended, however, is the velocity acquired after a very short time if the body starts from rest. This velocity is propor tional to the weight and can serve as a measure of the gravity. 2 DUHEM, JStudes sur Leonard de Vinci, Series II, p. 90. THE XlVth CENTURY 59 of the translation was never printed. At the beginning of the XVIth Century the remainder of the translation (Ethics, Politics and Economics) was published, together with Oresme's Treatise on the Sphere. In dynamics, Oresme was a disciple of Buridan and adopted from him the doctrine of impetus. Thus he maintains, in his Treatise on the Heavens and the World,1 written about 1377, that the acceleration of falling bodies is not, strictly speaking, accompanied by an increase of the heaviness of the body. Rather there is an increase of an " accidental property which is caused by the reinforcement of the isnelte (velocity) and this property can be called impetuosity (impetus). " This property is not the same as the heaviness " because if a hole were to be dug to the centre of the earth and then out the other side, and a heavy thing were to fall in this hole, when it came to the centre it would pass beyond it and rise, through the agency of this acquired and accidental property. Then it would fall back and go and come in the way that we see in a heavy thing hanging by a long string. There fore this is not strictly heaviness, since it is able to make [the body] ascend. " 7. ORESME'S RULE IN KINEMATICS. (UNIFORMLY ACCELERATED MOTION.) Oresme was above all a mathematician, and in this capacity he emerges as Descartes' forerunner in the matter of the invention of co-ordinates. As will be seen, and as Moritz Cantor has pointed out,2 we shall not stray from, the subject in hand if we emphasise this aspect of his work. We shall follow the Tractatus de Jiguratione potentiarurn et mensu- rarum difformitatum^ Oresme starts from the principle that every measurable thing can be thought of as a continuous quantity. Each intensity can be represented by means of a straight line erected vertical from each point of the " subject " which affects the intensity. Extension (longitudo) ia represented diagrammatically by a horizon tal line drawn in the direction of the subject. At each point of this line a vertical is erected whose height (altitudo or latitudo) is proportional to the intensity (intensio) of the property at the point corresponding to the subject. Thus the triangle of figure 16 represents a uniformly deformed quality (uniformiter difformis) terminated at a value zero. The 1 Bibliothtque National^ Paris, frcnch collection, Ms, 1083. 2 Vorlesungen uber die Ceschichte der Math^matik, 2nd Ed., Vol. II, p, 129. (Leipzig, 1900.) 8 Bibliothdque National^ Paris, latin collection, Ms. 7371, Trans. DUHEM. 60 THE ORIGINS a) b) c) d) Fig. 16 rectangle 6 represents a uniform quality and the trapezium c a uniform- ely deformed quality terminated by certain values at one end and at the other. Any other quality is said to be deformably deformed — that is, non-uniformly deformed or non-uniformly varying (difformiter difformis) . Such a one can be represented in the same way by erecting a vertical proportional to the intensity from each point of the extension. Oresme pointed out explicitly that the scale of such a representation could be chosen at will. Therefore the same quality can be represented by diagrams whose verticals are in a given relation to each other. Thus Oresme understands that the same quality can be represented by a diagram which is, for example, either circular or elliptical. He then proceeds to a classification of deformities according to the direction of their concavity and according to whether they arc rational (circular) or not. In this way he was able to enumerate 62 different kinds of deformity. Oresme came very near to modern analytical geometry when he wrote, " A uniformly deformed quality is such that when any three points of the subject are given, the relation of the interval between the first and the second to the interval between the second and the third is the same as the relation of the excess of intensity of the first over the second to the excess intensity of the second over the third. " This statement expresses the relation between the co-ordinates of three points on a straight line explicitly. Oresme went further than this in envisaging superficial qualities — qualities which had two dimensions with respect to the subject and whose intensity must be represented by a normal to a plane surface which defines the extension. Similarly, he put the question of how a corporeal quality — one having three dimensions with respect to the sub ject — can be represented. This passage merits quotation. " A superficial quality is represented by a solid figure. Now a fourth dimension does not exist and it is impossible to conceive of one. Nevertheless a corporeal quality may be thought of as having a double corporeity. One in a real extension, through the effect of the extension of the subject, has a locus in all dimensions. But there is also another which is only imagined and which arises from the intensity of the qua- THE XlVth CENTURY 61 lity. This quality is repeated an infinite number of times by the multi tude of surfaces which may be traced with respect to the subject. " In kinematics, Oresme accepted Albert of Saxony's ideas but ex pressed them with the help of his graphical representation. Velocity is susceptible of a double extension, either in time or with respect to the subject. It can be uniform or deformed with respect to each of these two extensions. Further, Oresme defined the total quality or the measure of a quality which was linear (or superficial) with respect to the subject as the area (or the volume) of the diagram which represents it. It is clear that if time is taken as the variable of extension, the measure or total quality of the velocity of a uniform motion is equal to the distance travelled. Oresme did not confine himself to this 4i instance. He contemplated a suc cession of uniform motions in the following way. He divides the time t into proportional parts which form a geometrical progression with ratio — and the first term -. The velo- 2. . 2. city has intensity ni in the nth interval. Under these conditions Oresme states that the total distance travelled is equal to four times the first rectangle, that is 4 UH • Oresme stated the following general rule for uniformly deformed qua lities — " Omnis qualitas, sifuerit umformiter difformis, secundum gradum puncti medii ipsa cst tanta quanta qualitas cjusdem subjecti. " That is, any uniformly deformed quality lias the same total quality (meas ure) as if it were related to the subject with the value which it takes at the middle point. Oresme verifies this rule for a linear uniformly deformed quality that starts with an intensity AC at A and finishes with zero value at J3. If D is the centre of the line AB which represents the 21 Si 1 t t t: t 2 4 6 16 Fig. 17 ° Fig. 18 B 1 It should be remarked in passing that this fact shows that Oresme knew how to calculate the sum of the series whose general term is— • 62 THE ORIGINS subject (subjectiva linea), the corresponding intensity is DE. The uniform quality that has the rectangle AFGB as its measure has the same measure as the uniformly deformed quality represented by the triangle ACB, because the area ACE is equal to the area AFGB. Oresme declares, "Any uniformly deformed quality or velocity is thus found to be equivalent to a uniform velocity, " but he does not explain, at this point, the identity of the measure and the distance travelled. So that he does not apply the rule which he has just for mulated to a uniformly deformed motion, although this rule includes the law of distance travelled in such a motion. Indeed, in modern lan guage we may write simply We have seen that some of the Schoolmen discussed the fall of bodies while others were concerned with kinematics, and that each of these sections developed a representation which contained a key to the law of the distance travelled by a moving body. But the union of these two problems was not effected. Undoubtedly the reason for this lies in the fact that the Schoolmen were satisfied when they had constructed abstract systems, whose niceties distracted their attention from the rudimentary experimental basis which they possessed. 8. ORESME AS A PREDECESSOR OF COPERNICUS. Except to the extent that it impinges on the history of the prin ciples of mechanics, we are not here concerned with the history of world- systems. However, by considering this aspect of Orcsme's nrnst original work, we shall see him to have been a prophet of Copernicus.1 The following quotations are taken from Nicole Oresrne'B Treatise on the Heavens and the World. Aristotle had established, in the second book of De Caelo? that the Earth remained motionless at the centre of the World* Grew me declared, " no observation could prove that the Heavens moved with a diurnal 2 motion, and that the Earth did not. " In this connection he made use, in an especially complete way, of the relativity of all motion. " If a man were placed in the Heavens, suppose that he were moved with a diurnal motion. Then if this man who is carried above the Earth sees the Earth clearly, and picks out the mountains, the valleys, 1 Cf. DUHEM, Revue gfafrale des Sciences, Nov. 15, 1909. 2 Lit., "journal." THE XlVth CENTURY 63 rivers, towns and castles, it will seem to him that that the Earth is moved diurnally just as, to us on the Earth, the Heavens seem to move. And similarily, if the Earth is moved with a diurnal motion and the Heavens not, it will seem to us that the Earth is still and that the Heavens move. " Oresme discussed Aristotle's argument according to which a stone thrown vertically upwards should fall to the west if the Earth is not at rest. In this connection, he declares that a stone thrown vertically in this way would be carried very rapidly towards the east, " together with the air through which it passes and with all the mass of the lower part of the World " which participates in the diurnal motion. In short, Oresme believed that the stone links its motion with that oft the Earth, from which it originally obtained its impetus. This thesis is correct, at least in its essentials.1 In a similar way, Oresme gave the lie to Ptolemy's argument that an arrow shot vertically from the deck of a ship moving very rapidly towards the east will fall far to the west of the ship. He then discussed the following reasons which had been given in support of the hypothesis that the Heavens moved and the Earth was stationary. 1) Any simple body can only have a simple motion. The Earth can only have a natural falling motion. 2) Apart from this natural falling motion could not have a circular motion — this motion, " which is violent, could not be perpetual. " 3) AverroSs holds that any motion from place to place can be related to a body at rest and, for this reason, he assumes that the Earth is necessarily fixed, at the centre of the Heavens. 4) All motion supposes a " motive virtue. " Now the Earth cannot be moved circularly by means of its heaviness. " And if it is moved in this way by some outside agency, such a motion will be violent and not perpetual. " 5) " If the Heavens were not moved with a diurnal motion, all Astrology 2 would be false. " 1 We now know that, in a free downward fall, a heavy moving body HufFers a small deflection towards the East as a result of the rotation of the Earth. The complete calculation requires that account he taken of the compound centrifugal forcet but a very simple intuitive argument can give the direction of this deflection, by means of the hypothesis that the motion of the body, starting from rest, proceeds according to the law of areas ^ ^ c<mgtant Initially, r === r0 and 6 = co, the velocity of rotation of the Earth. During. the motion r will decrease. The inequality r < r<> requires, by the law of areas, that 0 > to, which shows that the body is diverted towards the East. 2 Read " Astronomy. " 64 THE ORIGINS 6) Motion of the Earth would contradict the Holy Scriptures-- 44 Oritur sol et occidit, et ad locum suum revertitur . . . Deus firmavit orbem Terrae qui non commovebitur. " 7) The Scriptures also say that the Sun stopped in Joshua's time, and that is started its journey again in the time of King Hezekiah. And if it was the Earth which moved, not the Heavens, " such a cess ation would have been in versed. " Oresme's replies to these arguments were the following. 1) It is more reasonable to believe that every simple body and part of the World, except for the Heavens, is activated by a rotational motion in its natural place. And that, if a part of such a body is dis placed from its place in the whole it will, if this is allowed, return there as directly as possible. 2) The rotational motion of the Earth is certainly a natural one, but parts of the Earth that are displaced from their accustomed position have a natural, ascending or descending, motion. 3) " Supposing that circular motion requires the presence of some other body at rest, " it does not follow that the body at rest should be inside the body which moves, for there is nothing at rest inside a grindstone "except a single mathematical point, which is not a body." 4) The virtue which produces the rotational motion of the lower part, of the World is the nature of this part. It is the same thing that also pro duces the motion of the Earth towards its natural place when it has been displaced from it, in the same way that iron is drawn towards a magnet. 5) All appearances, all conjunctions, oppositions, constellations and influences in the Heavens remain unchanged when it is supposed that the motion of the Heavens is apparent and the motion of the Earth is real. 6) The Holy Scriptures are consistent here, " in the manner of ordinary human speech, " to the same extent that they agree in many other places. Things are not " as the letter sounds. " Thus it is written that God covered the Heavens with clouds — " Qui opcrit ("aelttm nubibus. " Now, on the contrary, all the evidence shows that it LH the Heavens which cover the clouds. Here again the words indicate the appearance and not the truth. It is the same for the motion of the Earth and the Heavens. 7) The stopping of the Heavens in the time of Joshua was an illusion — in fact, it was the Earth which stopped, and which started its motion again, or accelerated, in Hezekiah's time. Oresme then gives us several " good reasons " intended to show that the Earth has a diurnal motion and the Heavens do not. THE XlVth CENTURY 65 Every thing which benefits from another thing must set itself to receive, by its own motion, the benefit which it obtains from the other. Thus each element is moved towards its natural place, where it is kept. On the other hand, the natural place does not move towards the element. From which it follows that the Earth and the elements on the Earth, which benefit from the heat and the influence of the Heavens, must arrange themselves by their own motion to duly receive this benefit ; " also, to speak familiarly, as something which is roasted at the fire receives the heat of the fire about itself because it is turned, and not because the fire is turned towards it. " It is natural that the motions of the simple bodies of the World should have the same direction. Now, according to the Astronomers, it is impossible that all these motions should take place from east to west. On the contrary, if it is assumed that the Earth moves from west to east then this will agree with the other motions, " the Moon in one month, the Sun in a year, Mars in about two years and similarly for the others. " In this way, the part of the Earth which is habitable will be at the top and on the left of the World. u And it is reasonable that human habitation should be found in the most noble place that there is in the Earth. " According to Aristotle, the most noble thing which is and can be has its perfection at rest. Terrestrial bodies are set in motion towards their natural place in order to rest there. We pray that God should give the dead rest — Requiem aetcrnam. . . . The Earth, the most com mon thing, is displaced more rapidly than air, the Moon or the stars. In the hypothesis of the stationary Earth, the velocities which must be assigned to the stars, because of their distance from the centre, are inadmissible. The constellation of the North Wind— Major Ursa— -does not turn round, with the chariot in front of the cattle, as it would if it partook of a diurnal motion. All the appearances can be saved by means of a minor change — the diurnal motion of the Earth, whose size is so small in comparison with the Heavens — and without demanding so many different and incredible processes that God and Nature would have created for no purpose. By this means, the introduction of a IXth sphere is also made unneces sary. When God accomplishes a miracle, he does so cc without changing the common course of nature, except to the extent that this must be done. '* Thus it is natural that the arrest of the Sun in Joshua's time, 66 THE ORIGINS and the start again in Hezekiah's time, should be the result of terrestrial motion alone. Oresme concluded that considerations such as these " are valuable for the defence of our Faith. " More astute than Galileo, and safe from the thunderbolts that were to be hurled at this thesis later, he was nominated Bishop of Lisieux in reward for his work. Since, as we have said, Oresme's Treatise on the Heavens and the World was never printed, it is very unlikely that his ideas on the diurnal motion of the Earth could have become available to Copernicus. Perhaps the reader will decide that we have devoted too much attention to this early philosopher. But we have seen the best and the worst of Oresme's arguments about the system of the World. We have felt the mood of the time, at once naive and acute, fantastic and serious, familiar and dogmatic. Of the originality of Oresme as a mathemati cian, and of the vigour of his penetrating thought, there is no doubt. The prejudices of the Schools and the accepted ideas of the time did not imprison him. In the field of mechanics he was one of the first to address himself to the great French public, or, as he said himself, more accurately, " to all men of free condition and noble intellect. " 9. THE OXFORD SCHOOL. In his study of the representation of qualities Oresme invoked the authority of certain veteres whom he did not name. It is reasonable to believe that the ancients who preceeded Oresme in the general study of forms were the logicians of the Oxford School. One of the most eminent masters of this school was William Hey tes- bury, or Hentisberus. It is said that Heytesbury was a fellow of Merton College in 1330, that he belonged to Queen's College about 1340 and that he was Chancellor of the University of Oxford in 1371, Primarily Heytesbury was a logician of the most acute kind. But he was also concerned with kinematics, and it is in this connection that he claims our attention. In his Regulae solvendi sophismata the follow ing rule is given without proof— when the velocity of a moving body increases with the time in such a way that it is uniformly deformed, in a given time the body travels the same path as if it had moved uni formly with the velocity acquired half way through this time. This is Oresme's rule applied specifically to distance. To set against this, Heytesbury supported Thomas Bradwardme*B opinion— referred to earlier— that the effective velocity of a rotating body was that of its most rapidly-moving point. THE XlVth CENTURY 67 The most remarkable feature of Heytesbury's work is the appearance, albeit shrouded in obscurity, of the concept of acceleration. This was unknown to the Paris School. In fact, in his treatise De Tribus praedicamentis, Heytesbury distin guished between the latitudo motus (velocity) and the velocitas intensionis vel remissionis motus whose value was the increase or the decrease of the former. This quantity corresponds to acceleration. " For a moving body which starts from rest there can be imagined a range of velocity which increases indefinitely. In the same way can be imagined a range of acceleration or of slowing down (latitudo inten sionis vel remissionis) according to which a body can accelerate or slow down its motion with an infinitely variable quickness or slowness. This second range is related to the range of motion (velocity) as the motion (velocity) is related to the magnitude (distance) that may be travelled in a continous manner. " 1 Through this obscure language we catch the first glimpse of quantities that have become familiar tools of our trade, the vector representing J O the distance travelled, S ; the velocity (vector derivative), — ; and the d*S acceleration (vector derivative of the velocity), ~-r%- at We must also refer to the Liber Calculationum. In order to avoid the labyrinths of the Oxford School, we shall confine ourselves to a mention of Swineshead (variously called Suincet, Suisset, Suisseth, . . . by the continental copyists). The tradition of the XVth and XVIth Centuries, on the publication of Liber Calculationum in 1488, 1498 and 1520, added the epithet " Calculator " to this name. The document is the most typical of the Oxford dialectic that is available to us, and in spite of the relentless attacks of the Humanists, it was very highly regarded until the XVII th Century. Thus Leibniz, writing to Wallis, could express his wish to see it republished. Unfortunately, this work has only been attributed to Swineshead in error. Duhem, that tireless investigator, found a manuscript2 of this work which goes back to the XlVth Century, and in which the copyist attributes the work to Ricardus of Ghlymi Eshedi. (This must refer to William Collingham, a Master of Arts of Oxford.) This treatise is concerned with the general theory of forms, and sophist discussions make up the essential part of it. We shall quote a single extract from Chapter XV, which is called De medio uniformiter difformi. I Venice Edition, 1494. II Biblioth&que Nationals, Paris, latin collection, Ms. 6558, 68 THE ORIGINS " If the motion of a body is uniformly accelerated and starts with a value zero, the body will travel three times further in the second half of the time than in the first half. " This is a direct corollary of the law of distances in uniformly varying motion. In this way then, in XlVth Century Oxford, the kinematics of uni formly varying motion was known and commonly taught. The English School has the merit of having stated the law of distances more precisely than the School at Paris — on the other hand, it seems to have neglected Oresme's remarkable representation of uniformly deformed qualities. Just as much at Oxford as at Paris, these developments in kinematics, and the related general study of properties in intensity and extension, had no influence on the study of the fall of bodies. The description of these phenomena remained completely qualitative. 10. THE TRADITION OF ALBERT OF SAXONY AND OF BlJRIDAN. The tradition of Albert of Saxony and of Buridan was preserved in France and Germany by Themon, a son of Jew who taught at Paris in 1350, and Marsile of Inghen, who became rector of Heidelberg in 1386 after having been at the University of Paris in 1379. It is noteworthy that Marsile modified Buridan's doctrine in a somewhat unfortunate way. Thus he held that the impetus was at first weak in those parts of a body that were not in contact with the motive agency, and that it was strengthened there as the whole impetus became uniformly distri buted throughout the moving body. We must also refer to Pierre d'Ailly (1330-1420) who was high master of the College of Navarre in 1384 and who added the following original items to Albert of Saxony's paradoxes. " Someone who owns a field adjoining another piece of land, mid who excavates his earth in such a way that the area of the cavity remains constant, is defrauding his neighbour. " If the Earth is cut by a plane surface whose centre is at the centre of the World, when water is poured on this plane it will tend to assume the form of a hemisphere. " In the second place, if the bottom of a pool is flat, this pool will certainly be deeper in the middle than at the sides. ..." Pierre d'Ailly also gave Roger Bacon's paradox which has been quoted above (p. 52). With such intellectual games did the Schoolmen of the XlVth Cen tury delight themselves. So alive was this tradition that it maintained itself for over two hundred years. CHAPTER FIVE XVth AND XVIth CENTURIES THE ITALIAN SCHOOL BLASIUS OF PARMA THE OXFORD TRADITION NICHOLAS OF CUES AND LEONARDO DA VINCI NICHOLAS COPERNICUS THE ITALIAN AND PARISIAN SCHOOLMEN OF THE XVIth CENTURY DOMINIC SOTO AND THE FALL OF BODIES 1. BLASIUS OF PARMA AND HIS TREATISE ON WEIGHTS. Blasius of Parma (Biagio Pelacani), who became a doctor at Padua in 1347, taught at Padua and Bologna. He went to Paris about 1405 and died at Parma in 1416. His Treatise on Weights is known to us through a copy made by Arnold of Brussels and dated 1476. This treatise derives from Jordanus' School and links up the idea ofgravitas secundum situm — a first principle of Xlllth Century statics — with the tendency of a heavy body to fall along a chord rather than along an arc of a circle, and thus to take the shortest path, in Aristotle's sense, to its natural place. Blasius of Parma observed that when a balance with equal arms supporting equal weights is moved away from the centre of the World, these weights will appear to become heavier. Indeed, the line along which each of the weights tends to fall makes an angle with the vertical through the point of support which is the more acute as the balance is the further away from the centre of the World. This embellishment adds no thing useful to the positive statics of the authors of De Ponderibus. In a general way, Blasius of Parma was a critic and a sceptic who was content to multiply the objections to his predecessors' theories. For example, he observed that it is necessary to take account of passive resistances, though the correctness of the propositions of statics depends 70 THE ORIGINS on the process of neglecting all resistances occasioned by the medium. In a very naive way Blasius of Parma attempted to take these resistances into consideration. Apart from his Treatise on Weights, Blasius of Parma also wrote Quaes- tiones super tractatu de latitudinibus formarum, which was printed in 1486. In it he appears as an unsophisticated commentator of Oresme's doctrine. Although a critic of no great originality, Blasius of Parma was one of the means by which the statics of the Xlllth Century and the kine matics of the XlVth Century were handed on to the Italian School, which was destined to dominate mechanics during the period that we are going to study. 2. THE ITALIAN TRADITION OF NICOLE ORESME AND THE OXFORD SCHOOL. Together with Blasius of Parma, we must refer to Ga^tan of Tiene. Like Blasius, this author taught at Padua and died there in 1465, and he is responsible for having preserved the tradition of William Heytes- bury and the " Calculator " in Italy. One by one, he annotated the sophisms of the Oxford School, and his work was printed in Venice in 1494, together with the works of Heytesbury. In particular, GaStan of Tiene emphasised the distinction between latitude* motus (velocity) and latitude intensionis motus (acceleration). In this way the Italian School explained, more clearly than Heytesbury had done, the fact that a uniformly deformed motion corresponds to a constant latitude intensionis motus — that is, to a constant acceleration ; and that a deformably deformed motion corresponds to a uniformly deformed latitude intensionis motus. Bernard Torni, a Florentian physician who died about 1500, carried on the work of Gaetan of Tiene, and published Annotata to Heytesbury's treatise which made frequent mention of the " Calculator. " He was equally enthusiastic about Oresme's analysis, though he was only con cerned with the arithmetical procedures contained in this work. John of Forli, who taught medicine at Padua about 1409 and died there in 1414, wrote a treatise De intensione et remissions formarum which was printed at Venice in 1496. In it he refuted W. Burley, rejected Oresme's rule for the evaluation of a uniformly deformed quality, and attempted to introduce into medicine a terminology which was inspired by the Oxford School. The Humanists, especially VivSs, made him their target. It may be inferred, as Duhem has remarked,1 " that thanks to 1 DUHEM, Etudes sur Leonard de Finci, Series III, p. 509. XVth AND XVIth CENTURIES 71 Nicole Oresme, William Heytesbury and the * Calculator \ at the middle of the Quattrocento the Italian masters were well- acquainted with all the laws of uniformly accelerated or uniformly retarded motion. But it seems that none of them was inspired to assume that the fall of bodies was uniformly accelerated or, for this reason, to apply these laws to that phenomenon. " 3. NICHOLAS OF CUES (1404-1464) AND THE DOCTRINE OF " IMPETUS IMPRESSUS. " Nicholas of Cues studied at Heidelberg from 1416, and later at Padua in 1424. On returning to Germany he devoted himself to theology and science. He became Bishop of Brixen (Tyrol) in 1450 and died at Todi (Umbria) on August llth, 1464, His works were published in three parts between 1500 and 1514, and later reprinted at Basle in 1575. Nicholas of Cues was primarily a metaphysician. In De docta ignorantia he maintained that it was impossible to accept the idea of absolute truth, and argued the identity of the absolute maximum and the absolute minimum, as well as the existence of an Universe at once finite and unlimited. In mechanics, Nicholas of Cues has left us the dialogues De ludo globi. He is concerned with a game where a hemisphere is thrown in such a way that it meets some pins which are arranged in a spiral. The problem is to explain the trajectory of the body. Further, in the dialogue De Possest^1 he concerned himself with the gyroscopic motion of a toy top. " A child takes up this dead toy, devoid of motion, and wishes to make it live. For this purpose, by a procedure which he has invented and which is the instrument of his intelligence, he impresses on the toy the permanence of the idea which he has conceived. By a motion of his hands which is at once straight and oblique, consisting simultaneously of a pressure and a traction, he impresses on it a motion which is, for a top, supernatural. Naturally the plaything would have no other motion than the downward motion common to all heavy bodies — the child gives it the opportunity to move circularly, like the Heavens. This motive spirit, imparted by the child, is invisibly present in the ma terial of the toy — the length of time for which it remains there depends on the impressive force which communicates this property. When this spirit ceases to animate the toy, it resumes its motion towards the centre, as at the beginning. Do we not have here an image of what happened when the Creator wanted to give the spirit of life to an inanimate body? " 1 Dialogus trilocutorius de Possest, translated into French by DUHEM. 72 THE ORIGINS The World-system accepted at that time assumed that the motion of the different celestial spheres was maintained by that of the outer most sphere, itself activated by a Prime Mover. Nicholas of Cues held that is was sufficient that the Creator should have imparted an impetus to the spheres at the beginning, and that the impetus would then be conserved indefinitely. Thus we come across the " chique- naude, " the fillip, of which Pascal talked in connection with Descartes. This was also the doctrine of the Parisian Schoolmen of the XlVth Century, of Buridan and Albert of Saxony — in these celestial bodies there is no influence which can corrupt the initial impetus. In a manner which, for Nicholas of Cues, is very precise, the rotational motion of any perfect sphere is a natural motion. The impression of impetus on a moving body is comparable with the creation of a soul in the body. Nicholas of Cues became one of the inspirations of Copernicus and of Kepler, as well as of Leonardo da Vinci. 4. LEONARDO DA VINCI'S CONTRIBUTION TO MECHANICS. In mechanics, Leonardo da Vinci cuts the figure of a gifted amateur. Though he had read and meditated upon the Schoolmen that preceded him, his bold imagination was not inhibited as theirs was. He tackled all kinds of problem, often with more faith than success. Frequently he returned to the same problem by very different paths, and did not scruple to contradict himself. Leonardo made no concessions to systematics. But it seems that the original ideas which he threw off throughout his manuscripts were taken over by more than one of his successors. His work in mechanics is quite ^ sn n unique, and the few pages which we are able to devote, in this book, to an attempt at analysing its objective content can only provide a feeble echo of the torrent of ideas which flowed from this " autodidacte l par excellence." a) Leonardo da Vinci's concept of lg* 19 moment, — Leonardo da Vinci grasped the idea of moment and applied it in a most complete way to a heavy body turning about a horizontal axis, a body which he described as being " convolutable. " Thus, for a lever nb turning about a point n, Leonardo stated the following rule. 1 Autodidacte = one who is self-taught. XVth AND XVIth CENTURIES 73 " The ratio of the distance (length) mn to the distance nb is such that it is also the ratio of the falling weight at d to the (same) weight at the position 6. " l That is, the effect of a heavy body suspended at d is the same as if the body were suspended from the arm of the horizontal lever, nm, that is obtained by projecting nd onto the horizontal nb. Leonardo called this arm of a horizontal lever, equivalent to the inclined arm rad, the " arm of the potential lever. " It would seem here that he had read the XHIth Century statists.2 b) The motion of a heavy body on an inclined plane. — This is one of the problems that captured Leonardo da Vinci's interest, and which evoked some rather strange arguments from him. Thus we find the following passage among his writings. Fig. 20 " A heavy spherical body will assume a motion which is all the more rapid as its contact with its resting place is further separated from the perpendicular through its central line. The more ab is shorter than ac, the more slowly the ball will fall along the line ac [than along the vertical at] . . . because, if p is the pole of the ball, the part m which is outside p would fall more rapidly if there were not that small resistance provided by the counterpoising of the part o. And if there were not this counterpoise, the ball would fall along the line ac more quickly if o divided into m more often. That is, if the part o divides into m one hundred times, and the part o is missing throughout the 1 Les Manuscrits de Leonard de Find, published by Ch. Paris, 1890, Ms. E, fol. 72. 2 See above p. 42. 74 THE ORIGINS rotation of the ball, this will fall more quickly on n by one hundredth of the ordinary time Iff is the pole at which the ball touches the plane, the greater the distance between n and p, the more rapid the ball's journey will be. " i Elsewhere Leonardo wrote on the same subject in the following terms. " On motion and weight. All heavy bodies seek to faU to the centre, and the most oblique opposition provides the smallest resistance. " If the weight is at A9 its true and direct resistance will be AB. But the pole is at the place where the circumference touches the earth, and the portion which is furthest outside the pole falls. If SX is the pole, it is clear that ST will weigh more than SR, from which it follows that the part ST falls, that it dominates over SI? and lifts it up, and then moves along the slope with fury. If the pole were at IV, the more often AN divided into AC, the more quickly the wheel would run along the slope than if it were at X. " 2 Fig. 21 I am reluctant to comment on these texts and to attribute to Leo nardo things that he did not intend. Certainly he, like the aristotelianB, did not differentiate between dynamics and statics. It is also true to say that he reproduced and repeated the law of powers which Aris totle had formulated (see above, page 20). Further, Duhem, arguing from the relation of velocities that Leonardo gave, believes that he im mediately applied this same relation to powers — that is, to the apparent weights of a given body on differently inclined planes — and that he arrived in this way at the accurate law which we now accept. 1 Ms. A, fol. 52. 2 Ibid., fol. 21. XVth AND XVIth CENTURIES 75 I believe that it is more accurate to take a view of this kind — that Leonardo only sketched the solution of a problem which his rich im agination had formulated, but that he never gave it a final form. It certainly seems that Leonardo was unaware of the solution of the same problem that the unknown author of Liber Jordani de ratione ponderis had given, and which was based on the single concept of gravitas secundum situm, considered as a first principle of the statics of heavy bodies.1 ' m Fig. 22 Leonardo has, moreover, the merit of having attempted to solve this same problem of the inclined plane by another method, one which was unknown to his predecessors. Thus he observes2 that a uniform heavy body which falls obliquely divides its weight into two different aspects, along the line be and along the line nm. But here again we are left in suspense — he does not carry out the resolution of the weight into its two components along be and nm (normal to be). c) Leonardo da Vinci and the reso lution of forces. — Leonardo asked him self how the weight of a heavy body, sup ported by two strings, was apportioned between these two. He was of the opinion that the weight of the body suspended at b was divided between the strings bd and ba as the ratio of the lengths ea and de. This guess contradicts the now classical rule of the parallelogram. However Leonardo used, at least implicitly, the following rule. " With respect to a point taken on one of the components of a force, the moment of the other component is equal to the moment of the total force with respect to the same point. " Fig. 23 1 See above, p. 43. * Ms. G, fol. 75. 76 THE ORIGINS Fig. 24 Thus, through the intermediary of the concept of moment, Leonardo arrived at the resolution of forces. Indeed, on different occasions he drew the figure opposite, in which the weight JV is hung from two strings CB, CA, which are equally inclined to the vertical through JV. He wrote — " The pole of the angular balance formed of AD and AF is A9 and its appendages are DN and FC. " As the angle of the string that carries the weight N at its centre increases, the length of its potential lever decreases and the length of the potential counter-lever which carries the weight increases. " This remains somewhat mysterious, but, like Duhem,1 one may believe that the ten sion of the string CB, and the weight IV, would maintain the rigid body formed of the two potential arms AB and AF in equili brium, if the body were able to turn about the point A. " A confusion of ideas poured from Leonardo's mind but, to a high degree, he lacked the power of discriminating between the true and the false. Also, as an inevitable consequence, a truth which might emerge from the surface of incomplete or false beliefs and become clear to him at one instant, was thrown back again, to await the future which would finally return it to the shore. " d) Leonardo and the Energy of moving bodies. — Leonardo was aware of Buridan's doctrine through the intermediary of Albert of Saxony, who had adopted it. Moreover, he had read Nicholas of Cues. Leonardo reconciled these doctrines in the following way. At the outset he defined a quantity called impeto, analogous to impetus in Buridan's sense — " Impeto is a virtue created by motion and transmitted from the motor to the moving body, which has as much motion as the impeto has life. " 2 Elsewhere he says, " Impeto is the impression of motion which is transmitted from the motor to the moving body. . . . All impression desires permanance, as is shown us by the similarity of the motion impressed on the body. " 3 Leonardo regarded the motion of a projectile as being separated 1 0. S., Vol. I, p. 181. 2 Ms. E, fol. 22. 3 Ms. G, fol. 73. XVth AND XVIth CENTURIES 77 into three phases. In the first the motion is purely violent and is effected as if the projectile had no mass and was subject only to the initial impeto. In the third period, the impeto has completely disappeared. The moving body has a purely natural motion under the sole influence of gravity. Between these two extreme phases Leonardo assumed the existence of an intermediate period in which the motion was mixed, part violent, part natural. This is the period of compound impeto. The following quotation will illustrate this idea. " A stone or other heavy thing, thrown with fury, changes the direction of its travel half way along its path. And if you are able to shoot a cross-bow for 200 yards, place yourself at a distance of 100 yards from a tower, aim at a point above the tower and shoot the arrow. You will see that 100 yards from the tower the arrow will be driven in perpendicularly. And if you find it thus, it is a sign that the arrow has finished its violent motion and has started the natural motion, that is, that being heavy, it falls freely towards the centre. " x Or better still— " On convolutory motion. A top which loses the power which the inequality of its weight has about the centre of its convolution because of the speed of this convolution, because of the effect of the impeto which dominates the body, is one which will never have that tendency to fall lower, which the inequality of the weight seeks to do, as long as the power of the body's motive impeto does not become less than the power of the inequality. " But when the power of the inequality surpasses the power of the impeto, then it becomes the centre of the motion of convolution and the body, brought to a recumbent position, expends the remainder of the aforesaid impeto about this centre. " And when the power of this inequality becomes equal to the power of the impeto, then the top is inclined obliquely, and the two powers struggle with each other in a compound motion, both moving in a wide circuit, until the centre of the second kind of convolution is established. In this the impeto expends its power. " 2 Following the example of Nicholas of Cues, Leonardo concerned himself with the " game of the sphere ." He wrote — " On compound impeto. A compound motion is one in which the impeto of the motor and the impeto of the moving body participate 1 Ms. A, fol. 4. 2 MH. K, fol. 50. 78 THE ORIGINS together, as in the motion FBC which is intermediate between two simple motions. One of these is close to the beginning of the motion and the other close to the end. But the first is determined solely by the motor, and the second only by the shape of the body. " On decomposed impeto. Decomposed impeto is associated with a moving body which has three kinds of impeto. Two of these arise from the motor and the third arises from the moving body. But the two that arise from the motor are the rectilinear motion due to the motor and the curved motion of the moving body, and are mixed together. The third is the simple motion of the moving body, which only tends to turn round with its centre of convexity in contact with the plane on which it turns and lies. " L Here da Vinci's imagination is given free rein. Our author becomes even more lyrical when he defines the forza. " As for the forza — I say that the forza is a spiritual quality, an invisible power which, by means of an external and accidental violence, is caused by the motion and introduced, fused, into the body ; so that this is enticed and forced away from its natural behaviour. The forza gives the body an active life of magical power, it constrains all created things to change shape and position, hurtles to its desired death and changes itself according to circumstances. Slowness makes it powerful and speed, weak — it is born of violence and dies in freedom. The stronger it is, the more quickly it consumes itself. It furiously drives away anything that opposes it until it is itself destroyed— it seeks to defeat and kill anything that opposes it and, once victorious, dies. It be comes more powerful when it meets great obstacles. Every thing willingly avoids its death. All things which are constrained constrain themselves. Nothing moves without it. A body in which it is born does not increase in weight or size. No motion that it creates is lasting. It grows in exertion and vanishes in rest. A body on which it is im pressed is no longer free. " 2 Or again, " I say that forza is a spiritual, incorporeal, invisible power which is created in bodies which, because of an accidental violence, are 1 Ms. E, fol. 35. 2 Ms. A, fol. 35. Fig. 25 XVth AND XVIth CENTURIES 79 in some other state that their natural being and rest, I have said spiritual because there is in this forza an active incorporeal life, and I have said invisible because bodies in which it is born change neither in weight nor in shape ; of short life, because it always seeks to overcome its cause, and having done so, to die. " 1 We shall attempt, without too much quotation, to indicate Leo nardo's ideas on forza, ideas which were inspired by the metaphysics of Nicholas of Cues. Forza can be born of the " expansion undergone by a tenuous body in one that is dense, like the multiplication of fire during the firing of cannons. " It can also be born of a deformation as in a cross-bow. Finally, one forza can engender another — this is the case of impact. Leonardo returned to a pythagorean doctrine according to which a heavy body that is detached from a star to which it belongs tends to return there, in order to reconstitute the completeness of the star. He contrasted weight with forza, saying that these oppose each other. " Weight is natural and seeks stability, then rest — forza seeks killing and death for itself. " Weight is indestructible. When a heavy body arrives on the ground it exerts a pressure on it, " and penetrates, from one support to another, to the centre of the World. " A weight embodies power, /orza, motion and impact at the same time. But the fall of a body is itself preceeded by an accidental ascent. To be precise, at the origins of all actions in mechanics there must be a prime mover. And Leonardo, seduced by metaphysics, concludes — all motion arises from the mind. Further comment on this adventurous thesis of Leonardo seems, to us, unnecessary — its qualities are more of poetry than of precision, of eloquence than solidity, more metaphysical than positive. e) Leonardo da Vinci and perpetual motion. — Leonardo denied the possibility of perpetual motion on, the grounds that forza continually expends itself. On the other hand, gravity seeks to produce equili brium, all motions which are set in train by gravity have rest as their ultimate end. f ) Leonardo and the Figure of the Earth. — Having read and medi tated Albert of Saxony, Leonardo wrote in connection with the figure of the Earth — " Every heavy body tends downwards, and things which are at a height will not remain there, but will all, in time, fall down. Thus, in time, the World will become spherical and in consequence, will be com- 1 Ms. B, fol. 63. 80 THE ORIGINS pletely covered with water. " l And, without hesitation, he adds, — the Earth will be uninhabitable. In Leonardo's belief, the seas exerted no pressure on the part of the globe which they covered. Quite the contrary. " A heavy body weighs more in a lighter medium. Therefore the Earth, that is covered by air is heavier than that covered by water. " 2 g) Leonardo da Vinci and the theory of centre of gravity. The flight of birds. — Leonardo considered two towers ABIQ, CDL/C in " continual uprightness, " erected parallel to each other from the bases AB, CD, on the Earth. He pre dicted that " the two towers will tumble down towards each other if their construction is continued above a cer tain height in each case. " Here is his argument. " Let the two verticals through B and C be pro duced in c continual straightness. ' If they cut one of the towers in GC and the other in jBF, it follows that these lines do not pass through the centre of gravity of the lengths of the towers. Therefore KLCG, a part of one tower, weighs more than the remainder, GCD, and of these unequal things, one will be dominant over the other, in such a way that, of necessity, the greatest weight of the tower will carry away all the opposite tower. And the other tower will do the same, in a way which is inverse to the first. " 3 To recapitulate, Leonardo asserted that the vertical from the centre of gravity should not pass outside the base. This is, implicitly, the now classical theorem of the polygon of sustentation, but it contains the error, common to all the Schoolmen, that the convergence of the ver ticals has not been neglected. In this connection, Leonardo almost goes as far as to suggest that a measurement of the distance apart of two verticals at the top and the bottom of a tower should be used to deduce the length of the Earth's radius.4 Fig. 26 1 Ms. F, foL 70. 2 Ibid., fol. 69. 3 Ibid., fol. 83. 4 DUHEM, 0. S., Vol. II, p. 81, XVth AND XVIth CENTURIES 81 Going over from statics to dynamics, Leonardo, guided by his bold and ubiquitous imagination, affirmed that " any heavy body moves towards the side on which it weighs more. , . . The heaviest parts of bodies which move in air become guides for their motion. " x He also wrote, " Every thing which moves on a perfectly plane ground in such a way that its pole is never found between parts of equal weight, never comes to rest. An example is provided by those who slide on ice, and who never stop if their parts do not become equidistant from their centres. " 2 In his Treatise on Painting, Leonardo applied the preceding ideas to the flight of birds. 66 Any body that moves by itself will do so with greater velocity if its centre of heaviness is further removed from its centre of support. " This is mentioned principally in connection with the motion of birds. These, without any clapping of wings or assistance from the wind, move themselves. And this occurs when the centres of their heaviness are displaced from the centres of their support, that is, away from the middle of the extension of their wings. Because, if the middle of the two wings is in front of or behind the middle, or the centre, of the heaviness of the whole bird, then the bird will carry its motion upwards or downwards [and this all the more so] as the centre of heavi ness is more distant from the middle of the wings. ..." h) Leonardo and the fall of bodies. — It was inevitable that Leonardo should have become interested in the fall of heavy bodies. After having hesitated for some time between the two laws of velocity that were mentioned by Albert of Saxony (see above, page 57), Leonardo declared himself entirely in favour of the correct law v = kt . To set against this, the content of the studies on the latitude of forms (Oresme, Heytesbury) completely escaped him. Throughout he believed that motion (moto) was proportional to velocity (vclocitas) and, in consequence, was mistaken about the law of distances. In this connection we shall confine ourselves to a single quotation. " On motion. A heavy body which falls freely acquires one unit of motion in each unit of time ; and one unit of velocity for each unit of motion. " Let us say that in the first unit of time it acquires one unit of velocity. In the second unit of time it will acquire two units of motion and two units of velocity, and so on in the way described above. " 3 1 Ms. Et fol. 57. 2 Ms. A, fol. 21. 3 Ms. M, fol. 45. 82 THE ORIGINS i) Leonardo's hydrostatics. — Like the ancients, Leonardo set out to explain now water could appear in springs at the tops of mountains. He wrote, " It must be that the cause which keeps blood at the top of a man's head is the same as that which keeps water at the tops of mountains. " Leonardo sought this mechanism in the nature of heat, " There are veins which thread throughout the body of the Earth. The heat of the Earth, distributed throughout this continuous body, keeps the water raised in these veins even at the highest summits. " l To be accurate, Albert of Saxony, in his commentary of the relevant parts of Aristotle's treatise Meteores, had already invoked the intervention of heat in this matter. Leonardo was more fortunate when he gave a complete formulation of the law of the flow of currents. " All motion of water of uniform breadth and surface is stronger at one place than at another according as the water is shallower there than at the other. " Leonardo also outlined a theory of hydraulic pumps in the writing Del moto e misura delVacqua in which a hint of Pascal's principle can be discovered.2 j) Leonardo da Vinci and the geocentric hypothesis. — On looking for it in Leonardo's writings, there can always be found evidence of the kind that Duhem indefatigably sought. Thus there is the following passage, which is aimed at the geocentric hypothesis. " , , . Why the Earth is not at the centre of the circle of the Sun nor at the centre of the World, but rather at the centre of its elements, which accompany it and with which it is united. " 3 5. NICHOLAS COPERNICUS (1472-1543). His SYSTEM OF THE WORLD AND HIS IDEAS ON ATTRACTION. In this book we can only discuss the different World-systems to the extent that they have had an influence on the development of mechanics. The copernican system that was, in the hands of Kepler and Newton, to play a fundamental part in the creation of dynamics, had no imme diate influence on the scientists of the Renaissance. On the whole, these remained faithful to aristotelian ideas. We remark, for example, that the Sorbonne in the XVIth Century remained closed to copernican ideas and continued to teach Ptolemy's system. Our attention, therefore, should only be held for a short time by 1 Ms. A, fol. 56. 2 Cf. DUHEM, Etudes sur Leonard de VincL Series I, t> 198 3 Ms. F, fol. 41. * XVth AND XVIth CENTURIES 83 Copernicus' ideas on dynamics and the circumstances which facilitated the copernican revolution. From Antiquity there had been writers whose opinions were similar to those of Copernicus. Philolaus of Crete (a disciple of Pythagoras), Nicete of Syracuse and Aristarchus of Samos had attributed to the Earth both a daily and an annual motion, circular and oblique, about the Sun. (There was also supposed to an invisible earth which was symmetrical with ours with respect to the Sun.) In the Middle Ages William of Ockham, Buridan and Albert of Saxony assumed that the Earth could have a rotational motion which was not necessarily identical with the apparent motion of the stars. Albert of Saxony was not alone in attributing the precession of the equinoxes to a slow displacement of the Earth. We have seen in detail how Nicole Oresme, who was certainly unknown to Copernicus, had defended the theories of a fixed Heaven and an Earth which had a diurnal motion. The appeal to the doctrine of impetus in Oresme's thesis, which was used to destroy that of Aristotle, is especially important. In the religious fiels the Church in the XHIth Century, tolerant because of its power, had the wisdom to brush aside the a priori questions which could be opposed to every doctrine that deviated from the geo centric hypothesis. As early as 1277 Etienne Tempier, Bishop of Paris, made the assumption that the question of whether the Heavens had a translation motion, or not, could be discussed. Thus the Church in the XHIth Century assumed that the study of world-systems could be pursued as a piece of contingent research. In fact, a century later, Nicole Oresme did not compromise his ecclesiastical career in any way by believing in the motion of the Earth. In the field of metaphysics, even the debate on the plurality of worlds helped the copernican revolution. In reaction against the pytha- gorean doctrine, Aristotle explicitly understood the term " Heavens " (OvQKv6$) in the sense of " All " or of " Universe. " The absolute fixity of the Earth and the perpetual rotation of the Heavens consti tuted a dogma of science. The Universe is unique and each body has a unique natural place, to which it returns of its own accord if it is violently displaced from it. Any other world which can exist must necessarily be made of the same elements as ours* To Aristotle, this meant that the co-existence of several worlds implied a contradiction. Beyond this Eight Sphere there can be neither space nor time. This thesis was to be attacked in the Xlllth Century on the very grounds of the omnipotence of God. Michael Scot (1230) was of the opinion that God could have created several worlds, but that Nature would not have been able to accommodate 84 THE ORIGINS them. Saint Thomas Aquinas attempted to reconcile Aristotle's doc trine with the principle of divine omnipotence — the creation of similar worlds would be superfluous, the creation of dissimilar worlds would detract from the perfection of each of them, for only the ensemble can be perfect. In 1277 the theologians of Paris, at the request of Etienne Tempier, condemned the anti-pluralist thesis. William of Ockham intervened in the same direction — he argued that identical elements could simultaneously be directed towards different places. Thus a fire at Oxford would not move towards the same place as if it had been lit at Paris. The direction of a natural motion could therefore depend on the initial position of the element. Albert of Saxony decided against the plurality of worlds except u in a super natural way, to the liking of God. " On the other hand, towards the end of the XVth Century Joannes Majoris asserted, in his De infinite, not only the plurality of worlds but the existence of an infinite number of worlds. These discussions in no way lessen the originality of Copernicus' work, but to a certain extent they explain why he ventured to present his thesis. Being by profession Canon of Thorn, he protected himself with certain cautious declarations. Thus, in dedicating his works to Pope Paul III, he wrote, " I have believed that I would be readily permitted to examine whether, in supposing the motion of the Earth, something more conclusive (firmiores demonstrations) might not be found in the motion of the celestial bodies. " Strictly speaking, the doctrinal opposition only came much later with, for example, Melanchton and Father Riccioli. The latter was able to enumerate 77 arguments against the motion of the Earth and to refute 49 of the copernican arguments. As far as the Congregation of Cardinal Inquisitors was concerned, it only officially condemned Coper nicus' writings on March 5th, 1616. In order to fix certain essential dates, we recall that Copernicus was born at Thorn on January 19th, 1472. He received his doctorate at Krakov, and made his way to Bologna and then to Rome, where he devoted himself to astronomy. Copernicus gave himself up to a thorough study of the different world-systems which had been proposed by the Ancients, and used the motions of Mercury and Venus in order to place the Sun at the centre of the planets. In referring to the Pythagoreans, he proposed that the Sun should be placed at the centre of the World. Not wishing to ad vance anything without evidence, he started observation of planetary motions. The account of this task, completed in 1530, was only printed on his death in 1543, XVth AND XVIth CENTURIES 85 If, in not making the centre of the Earth coincide with that of the Universe, he dispensed with the aristotelian doctrine on an essential point, he kept, for the rest, most of the ideas of the Schoolmen. How ever, he did dispose of the distinction which Albert of Saxony had made between the centre of gravity and the geometrical centre of the Earth. We shall quote from Copernicus' De revolutionibus orbium caelestium. " The Earth is spherical because, on all sides, it strives towards the centre. The element of the Earth is the heaviest of all, and all heavy bodies are carried towards it and seek its intimate centre. " To my mind, gravity is nothing else than a certain natural quality given to the parts of the Earth by the divine providence of He who made the Universe, in order that they should converge towards their unity and integrity, by uniting in the form of a globe. It is probable that this property also belongs to the Sun, the Moon and to the wandering lights so that these too, by its virtue, keep that round shape in which we see them. " And here Copernicus attacks Albert of Saxony. " Because of their gravity, water and earth both tend towards the same centre. . . . One should not heed the Aristotelians when they claim that the centre of gravity is separate from the geometrical centre. . . . It is clear that both earth and water strive towards a unique centre of gravity at the same time, and that this centre is in no way different from the centre of the Earth. " Copernicus' doctrine on the figure of the Earth agreed perfectly with all the geographical observations. More simple than that of Albert of Saxony — an abstraction founded on prejudices opposed to the motion of the Earth — it was destined to triumph. But at this point the copernican ideas came up against a scholastic tradition whose root must be found in Aristotle's Meteores — the four elements earth, water, air and fire have equal masses and therefore occupy volumes which are inversely proportional to their densities. Moreover, the Aristotelians held that when a given mass of an element became " corrupted ** in order to produce the next element in the succession, its volume increased. Aristotle himself mentioned this relationship for the single instance of the transformation of water into air, but his commentators applied it without hesitation to the transformation of earth into water and, carrying the argument to the limit, said that the total volume of water was greater than the total volume of earth. Gregory Reisch, prior of Fribourg, put forward opinions of this kind in his Margarita philosophica (1496), a small encyclopedia that was widely circulated in the XVIth, Century. Twelve years after the end of Magellan's navigation of the globe — which should have clarified the scholastic opinion of the face 86 THE ORIGINS of the Earth — Mauro of Florence (1493-1566) took up Reisch's thesis again, and held that the volume of the closed earth was ten times less than that of the waters. Copernicus felt himself obliged to refute this author. We have only stressed this geophysical issue in order to show the kind of objection which the great reformer of the system of the world met during his lifetime. 6. JOHN FERNEL (1497-1558) AND THE FIGURE OF THE EARTH. John Fernel, chief physician to Henry II, deserves to be mentioned in a history of mechanics for having been the first among the moderns who had the initiative to measure a degree of terrestrial meridian. This he did by counting the number of revolutions of the wheels of his car riage between Paris and Amiens. In his Cosmotheoria, published at Paris in 1528, Jean Fernel disputed Albert of Saxony's doctrine, and decided in favour of the existence of a unique spherical surface for the combined mass of earth and water. He imagined the Earth to be abso lutely immobile and to present the shape of a globe which had been hollowed out in places and whose cavities had been filled with water. If one is to believe the chronicles,1 John Fernel, who was a distin guished astronomer and mathematician, would certainly have written other things " if his wife had not compelled him, so to speak, to leave the sterile study of mathematics. " 7. ITALIAN SCHOLASTICISM IN THE xvith CENTURY. At the beginning of the XVIth Century the Italian Schoolmen were divided into three camps ; there were the Averroists, the Alexandrists — those who made appeal to Alexander of Aphrodisias — and the Huma nists. As an example of the first school, we shall cite Agostino Nifo, to whom we owe a commentary of De Caelo et Mundo dated 1514. In this manuscript there is nothing but scorn for the parisian school of the XlVth Century, whose representatives are called Juniores, terminalista (nominalists), Sorticoles (disciples of Sortes, that is, of Socrates) and Captiunculatores (a corruption of Calculators). Albert of Saxony is ridiculed with the title of Albertutius or Albertus Parvus. The Averrolsts rejected the doctrine of impetus and returned to Aristotle's explanation of the motion of projectiles. Concerning the 1 LALANDE, Astronomic, Vol. 1, p. 189. XVth AND XVIth CENTURIES 87 fall of heavy bodies, Nifo, like Saint Thomas Aquinas,1 held that proxi mity to its natural place contributed towards a body's acceleration. He added to this an " instrumental cause, " and a quality belonging to the moving body. Among the Alexandrists Peter Pomponazzi of Mantua, who is said to have taught at Bologna, devoted a treatise De intensione et rernissione formarum (1514) to an attack on the Oxford School. In his De reactione (1515) he called William Heytesbury " the greatest of the Sophists " and contrasted him with " the clear and great voice of Aristotle. " The thesis of Alexander of Aphrodisias, to whom this faction gave allegiance, has been preserved for us by Simplicius. It consisted of the assumption that a heavy body which was placed at a height became lighter. This lightness obtained at the beginning of the fall and then, continuously, became less apparent. The Italian Humanists reproached the Schoolmen for their " parisian manner, " which was described as barbarous, sordid, gross and uncul tured, but approved of their religious orthodoxy. In addition the Humanists, in the person of Giorgio Valla who taught at Padua (1470) and at Venice (1481), made an especial attack on the AverroXsts. These were taken to task for their language — studded with arabic terms — and for their exclusive cult of Aristotle and consequent neglect of Plato. Valla went as far as to consider Averroes, in Latin, of course, as a " primitive creature emerging from the mud " and as " pigheaded. " In dynamics Valla echoed the thesis of intermediate rest (quies inter" media) between the ascent and the descent of a body, which compro mised the continuity of the motion. He assumed the existence, in every moving body, of a vis insita* This quantity, however, has no connection with Buridan's impetus, but rather is accounted for by the proximity of a motive agency or the natural place, according as a violent or a natural motion is in question. These violent polemics added nothing new to mechanics, and we have only described them in order to illustrate the atmosphere of the time, with which original thinkers had to contend. 8. PARISIAN SCHOLASTICISM IN THE xvith CENTURY. The teachings of Buridan and Albert of Saxony were preserved at the College of Montaigu under the Scotsmen Joannes Majoris and George Lockhart. Jean Dullaert de Gand and the Spaniard Luiz Coronel taught at the same college. Another Spaniard, Jean de Celaya, taught at Sainte-Barbe. 1 See above, p. 57. 0« THE ORIGINS This tradition was eclectic. It declared that it followed " the triple voice of Saint Thomas Aquinas, the Realists and the Nominalists. " Nevertheless, this School quibbled and argued much more than the masters of the XlVth Century, and lacked their originality. Joannes Majoris, who was primarily a great teacher, taught Oresme's work on the latitude of forms and, in 1504, had Buridan's Summulae printed. In his Disputationes Theologiae Majoris argued, more explicitly than Buridan had dared to do, the identity of the dynamics of celestial and terrestrial bodies. Thus, like Nicholas of Cues, he prepared the way for Kepler. But it was left to him to fight the Reformation and to defend the dialectic which the students had begun to neglect. It was a time when, " covered with threadbare garments and with empty purses, the unhappy logicians of the University of Paris mused sadly on chairs which were no longer surrounded by pupils. They listened to the raillery that was poured on their learning, which they had only acquired with great effort, and to which they had consecrated their working lives. " l Already attacked by the Humanists, Scholasticism no longer paid. In 1509 Jean Dullaert de Gand (1471-1513) continued the printing of Buridan's works. In 1506 he himself published some Quaestions on Aristotle's De Caelo and Physics. At Montaigu he taught the doctrine of impetus. He assumed that the impetus was modified by the shape of the projectile and supported the notion of intermediate rest, which he took to be at the moment when the impetus of the ascending motion was overcome by the gravity. He came to no conclusion as to the nature of impetus, whether it was a distinct property of a moving body or not. Concerning the fall of heavy bodies, he assumed that the im petus increased continually, though he did not know whether it should be taken as proportional to the size of the body. Similarity, Dullaert taught Oresme's rule on uniformly deformed motion through the reading of Bernard Torni, though he confined the treatment to the algebraic form of the rule. He lost himself in discussions on the nature of motion, a " successive entity truly distinct from all permanent things. " Luiz Nunez Coronel, of Segovia, published Physicae Perscrutationes in 1511. In dynamics, he believed in the gradual weakening of the impetus of all violent motion, which was a serious regression from Bu ridan's thesis. As for intermediate rest, he " imagined instances in which a stone thrown in the air remains there at rest for as much as an hour, two hours or even three, " without being dismayed by the objection that such rest was never seen. " This objection is not conclusive. The 1 DUHEM, Etudes sur Leonard de Vinci, Series III, p. 179. XVth AND XVIth CENTURIES 89 great distances may prevent one from seeing the rest, or it may even happen that the stone remains motionless for a time which is imper ceptible. " To Coronel, impetus was an aptitude of the moving body, a certain " actual entity, " produced in it by means of a repeated series of local motions. Impetus was thus identified with a cognition acquired by the repetition of the same perception, like that of handwriting to the fingers of the hand. This physiological model, however arbitrary it may be, was taken up again by Kepler. In the theory of gravitation, Coronel showed himself to be singularity naive. If weight is a property emanat ing from the natural place, in order to prevent this property from passing through the surface of the earth, it will be sufficient if this is covered with a garment. . . . Elsewhere Coronel attributes the generation, in a free fall, of an impetus of greater or lesser intensity, exclusively to gravitation or to the substantial form of the heavy body. In the motion of projectiles, Coronel assumed a mixture of decreasing impetus and progressive agitation of the air, which resulted in a certain compensation and assured a maximum violence at the middle of the trajectory. In 1517 Jean de Celaya published Expositio in libris Physicorum, a literal commentary on Aristotle. The relevant discussion only appear ed later under the title Sequitur glosa, and is distinguished by having explained, rather clearly, a law of inertia in the following terms. " It would follow from the theory that a body which is projected will move forever. However, this result is false and the reason is clear. The theory does not include anything which will destroy the impetus, and it will therefore move the projectile forever. u To this we reply by refusing to recognise the validity of the argu ment, and this because we deny the antecedent. Indeed, this impetus is sometimes destroyed by the resisting medium, sometimes by the shape or the property of the projectile that exerts a resisting action, sometimes, finally, by an obstacle. " Celaya assumes that in the absence of these three mechanisms of destruction, impetus lasts indefinitely. " It is not necessary to suppose as many intelligences as there are heavenly bodies. It is sufficient to say that there is in each star an impetus? that this impetus was put there by the Prime Cause, and that it is this which moves the star. This impetus is not modified for the very reason that the heavenly body has no inclination towards a different motion* " This is entirely in agreement with Buridan's thesis. In the general sense, impetus was a second quality to Celaya. He compared it to ** knowledge and dispositions of the soul. ** 90 THE ORIGINS Celaya was rather reticent on the subject of the plurality of worlds. The Catholic faith provides no argument from which the existence of several worlds can be deduced, and the Philosopher (Aristotle) saw objections to such a happening. All the same, " from the supernatural point of view, there can exist several worlds, either simultaneously or successively, either concentrically or excentrically. " For " God can do all things that do not imply a contradiction, " and here there is none. The opinion of the Philosopher according to which the World contains all possible matter " is heretical, and the Philosopher would not be able to prove it. " Finally, we remark that though Celaya taught the work of Nicole Oresme and the Oxford School, like Jean Dullaert, he only knew them through the Italian tradition. 9. THE ATTACK OF THE HUMANISTS. The rapid sketch which we have given above is sufficient to show that Parisian Scholasticism in the XVIth Century was in regression from the original work of the XlVth Century. The Humanists who were to proclaim its decadence were pupils of the College of Montaigu — Didier Erasme and Jean Luiz Vives. Moreover, in mechanics, these Humanists preserved the tradition which they had received from Majoris and Dullaert. Thus, in his immensely successful Colloquia (1522), Erasme discussed the oscillation of a heavy body that travelled through to the centre of the Earth — this is, as we know, a problem that had already been raised by Oresme — in terms that his masters would not have disowned. Erasme's Eulogy of madness, which antedates the Colloquia and was published in 1508, includes a determined attack on the theologians, " those quibblers who are so puffed up with the wind and smoke of their empty and quite verbal learning that they will not give way on any point. " Jean Luiz Vives (1492-1540) was born at Valence and was a pupil of Jean Dullaert before becoming a professor himself at Louvain. In De prima philosophia (1531) he discussed " intermediate rest " at great length, in terms which were in complete conformity with the pure scho lastic doctrine. He had therefore retained traces of the teaching of Montaigu. His violent diatribes were directed at the Parisian masters and at the Oxford School with its XVth Century tradition, which sought to extend the Calculator's dialectic to medicine. In De philosophiae naturae corruptione (1531) Vives wrote, " How can there be learning in subjects so divorced, so completely separated, from God on the one hand and from sensibility and spirit on the other ? XV th AND XVIth CENTURIES 91 In a domain in which, founded on nothing, there is seen a vast structure of contradictory assertions concerning the increase and decrease of intensity, the dense and the tenuous, uniform motion, non-uniform motion, uniformly varying motion and non-uniformly varying motion ? It is not possible to count those who, without any limit, discuss instances which never occur, which could never turn up in nature ; who talk of infinitely tenuous and infinitely dense bodies ; who divide an hour into proportional parts for this reason or that, and consider, in each of these parts, a motion, or an acceleration, or a rarefaction, varying in a given way. . . . " Further, in De medicina we find, ** the young people and adolescents who have been educated by means of these specious and tricky discus sions know nothing of plants, of animals, nor of nature in the round. They have been brought up with no experience of natural things, without knowledge of reality. They have no prudence. Their judge ment and their counsel are excessively weak, and yet they are expected to be able to win honour for themselves ! " And he concludes ( In pseudodialecticos) , " For myself, I have a great gratitude to God, and I thank him that I have at last left Paris, that I have emerged from the Cimmerian darkness, have come out into the light, that I have discovered the truly dignified studies of mankind — those which have earned the name Humanities. " 10. DOMINIC DE SOTO (1494-1560) AND THE LAWS OF FALLING BODIES. At the very moment that Scholasticism appeared to be discredited by the attacks of the Humanists, there intervened an original work which succeeded in formulating the laws of falling bodies correctly. We shall now analyse this work in some detail. Dominic de Soto was born in 1494, the son of a gardener at Segovia. He attended the University of Alcala of Henares, and then took himself to the University of Paris where the Spaniards were already rather numerous. He returned to Alcala in 1520 and gave up the chair which he had obtained in order to take the habit of a preaching friar. From 1532 to 1548 he taught theology at Salamanca. As confessor to Charles V, he followed his king to Germany. Later he returned to Salamanca and taught theology there from. 1550 until his death in 1560. Soto had been a witness of the furious attacks of the Humanists upon the Paris School but remained, for his part, a Schoolman. However, he eschewed nominalism and attacked it in his Quaestiones (1545) on Aristotle's Physics. 92 THE ORIGINS We shall not discuss the metaphysical content of Soto's work, in which he rejected the concept of an actual infinity in favour of a virtual one, and shall only be concerned with his contribution to mechanics. In the first place we note, in passing, that Soto adopted Albert of Saxony's opinion on the equilibrium of the earth and the seas. In connection with the motion of projectiles, he taught the doctrine of impetus, and presented it in the following way. " First Conclusion. — It cannot be denied that a man or a mechan ism sets the air in motion when throwing a projectile, just as we see the circular agitation of water around a stone which has been thrown into it. The truth of this conclusion is especially evident for cannons, from which the air is driven in the form of a very violent explosion at the same time as the shot. " Second Conclusion. — Air is not the only cause of the motion of projectiles. Whatever has thrown the moving body is also a cause, through the intermediary of the impetus which it has impressed on the body. " i Thus Soto sought to reconcile Aristotle's doctrine with that of impetus by assuming that the agitation of the air played some part in the motion of projectiles. However, he summarily dismissed Marsile Inghen's opinion (see above, page 68). " Observation proves that air too is a cause of the motion of pro jectiles. Indeed, we know that an arrow does not hit an object which is near with as much violence as it hits one that is a little more distant. This is why Aristotle says, in the second book of the Heavens, that natural motion is more intense towards the end, while the greatest intensity of the motion of a projectile is attained neither at the beginn ing nor at the end, but near the centre. " Some suppose that the reason for this happening is the following one — the impetus is not all imparted to the arrow at the first instant. Later it becomes more intense, or else distributed through the extension of the arrow, so that it moves it in a more urgent way. But this is not very easy to understand. Indeed, one cannot see what could increase the intensity of the impetus after the arrow has been separated from the ballista, for an accident does not, of itself, become more intense. On the other hand, as the arrow is a continuous body, the impetus is simul taneously imparted to the whole body. Therefore it cannot distribute itself further later. " 2 1 Quaestiones in libros Physicorum, Vol. II, fol. 100. 2 Ibid. XVth AND XVIth CENTURIES 93 Soto regarded impetus as a " property distinct from, the subject in which, it is encountered, " like gravity or lightness. Conversely, he saw gravity as a " natural impetus. " In his desire to reconcile Aristotle and Buridan, Soto went as far as to argue that Aristotle did not doubt the doctrine of impetus, but that he must have taken it as obvious, from the analogy with heavy and light bodies, and passed over it in silence. But the essential part of Soto's work is that which concerns the fall of bodies. Some of the Schoolmen who had proceeded him had discussed the fall of bodies, albeit in a purely qualitative manner; others had discussed uniformly varying motion in the field of pure kinematics ; but these studies had remained separate. It has now been established that the synthesis of these discussions was accomplished in Soto's time. He himself does not describe this achievement as a personal success. Is this modesty on his part or, on the other hand, the reflection of a move ment which had already been completed by the Schoolmen ? The answer to this question is of little importance — what does matter is the law which was clearly expressed by this Spanish master. We shall quote Soto's own text, as translated by Duhem.1 " Motion which is uniformly deformed with respect to time is that in which the deformity is so — if it is divided according to time, that is according to intervals which succeed each other in time, in each part the motion at the central point exceeds the weaker terminal motion in this part by an amount equal to that by which it itself exceeded by the more intense terminal motion, " This kind of motion is one which is appropriate to bodies which have a natural motion and to projectiles (Haec motus species proprie accidit naturaliter motis et projectis) . u Indeed, each time that a mass falls from the same height in a homogeneous medium, it moves more quickly at the end than at the beginning. On the contrary, the motion of bodies which are projected [upwards] is weaker at the end than at the beginning. And similarily the first motion is uniformly accelerated and the second, uniformly retarded. " Soto was concerned with the law of distances for uniformly varying motion, and in his writings the ideas of Nicole Oresme and of the Oxford School may be clearly identified. After some hesitation he declared himself for the correct law. " Uniformly deformed motion with respect to time follows almost the same law as uniform motion does. If two bodies travel equal 1 Theologi ordinis pr dedicator um super octo libri Physicorum Aristotelis Quaestiones, Salamanca, 1572, fol. 92 d. 94 THE ORIGINS distances in a given time, even though one moves uniformly and the other in any deformed manner — for example, in such a way that it covers one foot in the first half-hour and two feet during the second — from the moment that the latter covers as many feet as the former, which moves uniformly, in the whole hour, the two moving bodies will move equally. " But here an uncertainty arises. Should the velocity of a body in uniformly varying motion be denominated by its most intense degree ? If for example, the velocity of a falling body increases in one hour from degree zero to degree eight, should it be said that this body has a motion of degree eight ? It seems that the affirmative reply is the correct one, for this is the law which seems to be followed by uniformly varying motion with respect to a subject moving body. Nonetheless we reply that the velocity of uniformly varying motion is evaluated by the mean degree and should be given the denomination of that degree. One should not argue in this respect as in the case of uniformly varying motion with respect to the subject. Indeed, in the latter case the reason for the rule adopted is the following — each part of the moving body describes the same line as the most rapidly moving point, in such a way that the whole moves as quickly as this point. Whereas a body which moves with a motion that is uniformly deformed with respect to time does not describe a path as great as if it were mov ing uniformly with the velocity which it attains at its supreme degree. This goes without saying. Therefore we believe that uniformly de formed motion should be denominated by its mean degree. Example — // the moving body A moves for one hour and constantly accelerates its motion from degree zero to degree eight, it will travel just as great a path as the moving body B which moves uniformly with degree four for the same time. " It follows from this that when bodies move with a deformed motion, these motions should be reduced to uniform ones. " 1 1 Ibid., fol. 93 and 94. CHAPTER SIX XVIth CENTURY (Continued) THE ITALIAN SCHOOL OF NICHOLAS TARTAGLIA AND BERNARDINO BALDI 1. NICHOLAS TARTAGLIA. Nicholas Fontana, called Tartaglia, derived his surname — which indicates stammering — from an injury obtained when he was wounded, while still an infant, in the sack of Brescia. He was born at Brescia at the beginning of the XVIth Century and died at Venice in 1557. Tartaglia was one of the means by which the original statics of the XHIth Century, which had been forgotten, was preserved for the Italian School of the XVIth Century. Indeed, Tartaglia entrusted Curtius Trojanus with the publishing of the work of the unknown author of the XIHth Century. This appeared in 1565 under the title of Jordani opusculum de ponderosite Nicolai Tartaleae studio correctum.1 Instead of giving his predecessors credit for their work, Tartaglia, who was not very scrupulous in matters of scientific pro priety, claimed their demonstrations as his own. Dynamics is treated in two of Tartaglia's works, Nova Scientia (1537) and Quesiti et inventioni diversi (1546). In the first of these works Tartaglia attributes the acceleration of falling bodies to their approach to their natural place. " A heavy body hastens towards its proper nest, which is the centre of the World, and if it comes from a place which is more distant from this centre it will travel more quickly in approaching it. " Elsewhere he distinguished three phases in the trajectory of a projectile — AB (rectilinear), BC (a curved join) and CD (vertical). He held that the velocity was Fig. 27 1 See above, pp. 41 to 46. 96 THE ORIGINS least at C, at the point at which the violent motion finished and the natural motion began. This dynamics is improved a little in the Quesiti, in which he asserts that, except for the case in which the particle is thrown vertically, the trajectory of a shot has no rectilinear portion. It is the natural gravity which makes the trajectory curve downwards. The more rapidly a heavy body is thrown in the air, the less heavy it is and the straighter it travels through the air, which supports a lighter body more effectively. The more the velocity decreases the more the gravity increases, and this gravity continually acts upon the body and draws it towards the earth.1 Tartaglia adds that the motion of a projectile starts with an accel eration. He says that, for the same cannon with the same charge of powder and the same elevation, a second shot will go further than the first because it will find the air already divided and more easily penetrable. 2. JEROME CARDAN (1501-1576). Jerome Cardan was born at Padua in 1501, died at Rome in 1576, and was at once physician, astrologer, algebraist and a student of mechanics. Cardan's two works which are relevant to mechanics are the De Subtilitate (1551) — which was translated into French by Richard le Blanc in 1556 — and the Opus novum (1570). In statics Cardan believed that that he had surpassed Archimedes, whom he had read and admired, by treating the weight of the two arms of a balance. Indeed he wrote, " The heavinesses [moments] of the two arms of a beam [horizontal, cylindrical and homogeneous] have the same proportion to each other as that of the squares of the lengths of the two arms . . . Hoc est quod Archimedes reliquit intactum. " 2 Cardan uses the concept of moment fully. " It is clear that, in balances and in things which lift loads, the further the burden is from the fulcrum the heavier it is. Now the weight at C is separated from the fulcrum by the length of the line CJ5 and that at JP, Fig. 28 by the length of the line FP. " 3 1 C/. DUHEM, Etudes sur Leonard de Vinci, Series III, p. 188. It may be that Tartaglia used Leonardo da Vinci's notes without acknowledgement. Opus novum, Proposition XCIL 3 De subtilitate, translated by RICHARD LE BLANC, p. 16. XVIth CENTURY 97 Like his predecessors of^the Xlllth Century, Cardan then consi dered equal arcs FG and CE starting from the points jP and C, but he directed his attention to the velocities and not to the paths, by observing that the fall from F to G was more " tardy " than the fall from C to E. He concludes, " then this argument is general — that the further the weights are from the end, or the line fall of along the straight line or the oblique, that is to say along the angle, the heavier they are. . . . Thus the intention of the weight is to be carried directly towards the centre. But because it is prevented from doing this by the linkages, it moves as best it can. " l Duhem interprets this rather obscure passage in the following way. " When a heavy body falls vertically the power of the body is measured, as Aristotle intended, by the velocity with which it falls. But through the agency of the mechanism that carries it, because of the linkages or constraints — to use the modern term — it may happen that the body does not move vertically. Therefore in order to reckon its motive power it is necessary to take account, not of the body's total velocity, but only of the vertical component of this velocity, or in other words, of the velocity of fall. " If then a given weight is suspended from some point of a solid which can move about a horizontal axis, the power of this weight will be greater as the point of suspension falls more rapidly when a given rotation is applied to the support. Therefore it will be greater as the point of suspension is further from the vertical plane containing the axis. " 2 Like Leonardo, Cardan investigated the pul ley-block, together with the screw and the jack. Cardan was of the opinion that on an inclined plane the heaviness of a given body is proportional to the velocity with which it moves down the plane. Therefore this hea viness is zero on a horizontal plane and increases with the angle of inclination. Car dan assumes that the apparent weight is (, proportional to this angle. " Let a sphere a, of weight £, be placed at <lg" "" the point b and suppose that it is desired to draw it along the plane be. The vertical plane is bf. On the horizontal plane be the force needed to move a may be taken as small as desired. . . . 1 Ibid. 2 DUHEM, 0. S., Vol. I, p. 46. 98 THE ORIGINS Consequently, according to tke consensus of opinion, the force which will move a along be will be zero. On the other hand, a will be moved towards / by a constant force equal to g ; in the direction be by a constant force equal to fc ; in the direction bd by a constant force equal to ft. Since the motion along be is produced by a zero force, the rela tion of g to k will be as the relation of the force which moves a along bf to the jforce which moves a along 6c, and as the relation of the right angle ebf to the angle ebc. In the same way the force which moves a along bf isjx> the force which moves a along bd as the angle ebf is to the angle ebd. " l There is no clearer distinction between statics and dynamics in Cardan's work than can be discovered in that of Leonardo da Vinci. Like Leonardo, Cardan asserted the impossibility of perpetual motion unless natural motions were in question. We shall quote De Subtilitate on this subject. " Either the continuity of motion will arise from the fact that the motion is in conformity with nature, " (hereby Cardan excepts the motion of the Heavens), " or else this continuity will not be maintained equal to itself. Now that which continually diminishes and is not augmented by some external action, cannot be per petual. . . . " The motions that bodies can have are of three kinds ; they may essentially tend to the centre of the World ; they may not be directed towards the centre in a simple way, like the running of water ; or they may stem from a particular characteristic, like the motion of iron to wards a magnet. Patently, perpetual motion should be sought in mo tions of the first two kinds. Now when a weight is pulled more strongly, or held back more energetically, than is consistent with its nature its motion is natural, it is true, but not free of violence. Examples of these two conditions are seen in the weights of clocks. As for motion in a circle, this only belongs naturally to the sky and the air, and the latter is not actuated by an ever-present mechanism. For other bodies, it [motion in a circle] always has its root in vertical motion. Thus in rivers, at the rate and to the extent that the waters are generated by the source, they continually descend along the slope of the bed. Now in order that a motion should be perpetual, it would be necessary that the bodies which were displaced and came to the end of their path should be carried back to their initial position. But they can only be carried there by means of a certain excess of motive power. ..." 1 Opus novum. Proposition LXXIL XVIth CENTURY 99 3. JULIUS-CAESAR SCALIGER AND BURIDAN'S DOCTRINE. Julius-Caesar Scaliger was a supporter of the parisian Scholasticism and one of Cardan's opponents. The latter, in Book XVI of De Subti- litate, had had the naive audacity to make a classification of genius, in order of decreasing merit, in the following way — Archimedes, Aristotle, Euclid, John Duns Scot, Swineshead the Calculator, Apollonius of Pergum, Archytas of Tarento, etc. . . . Scaliger replied on this matter. " You have given a simple artisan the place above Aristotle, who was not less erudite than he in these same mechanical skills ; above John Duns Scot, who was like the file of truth ; above Swineshead the Calcu lator, who almost surpassed the limits imposed on the human intelli gence ! You have passed over Ockham in silence, that genius who outwitted all previous geniuses. . . . You have placed Euclid after Archimedes, the torch after the lantern. . . . " x Scaliger explicitly refused to consider the agitated air as the seat of the motive agency of projectiles, and accepted Buridan's doctrine in all but form. In this he differed from Cardan, who remained an Aristo telian in this matter and who added nothing to the work of Tartaglia and da Vinci. " The motio (here synonymous with impetus) is an entity which implanted in the moving body and which can remain there even when the prime mover is taken away. By prime mover I mean that which causes this entity to penetrate into the body. For it is not necessary that the efficient cause should continue to exist with its effect. " 2 Scaliger continued — " Heavy bodies, stones for example, have nothing which favours their being set in motion. They are, on the contrary, quite opposed to it. ... Why then does a stone move more easily after the motion has started ? Because the stone has already received the impression of motion. To a first part of the motion a second succeeds, and each time the first remains. So that, rather than a single motor exerting its action, the motions which it imparts in this continuous succession are multiplied. For the first impetus is kept by the second, and the second by the third " 3 4. BENTO PEREIRA (1535-1610). THE CLASSICAL REACTION. In 1562 Bento Pereira published at Rome a treatise called De com- munibus omnium rerum naturalium principiis which became very po- 1 Exotericarum evercitationum libri, Paris 1 557t Exerc. 324. Translated by DUHEM. 2 Ibid., Exerc. 76. 3 Ibid., Exerc. 77. 100 THE ORIGINS pillar and which was studied by Galileo himself. Bento Pereira knew of Scaliger's Exercitationes but adhered, himself, to Aristotle's doctrine on the motion of projectiles. Cesalpin and Borro may also be cited as representatives of this classical reaction. 5. THE " MECHANICORUM LIBER " OF GUIDO UBALDO (1545-1607). We now come to a student of mechanics who was a great authority until the beginning of the XVIIIth Century, and who was one of Galileo's masters. Descartes, who gave few references, recalled having read him and even Lagrange quoted him often in the historical part of his Meca- nique Analytique. To the classical impedimenta of the Medieval authors, Guido Ubaldo added a reading of Archimedes and of Pappus, and through the latter achieved a partial knowledge of Hero of Alexandria. His Mechanicorum Liber is dated 1577. Guido Ubaldo, who was Marquis del Monte, lived in seclusion in his Castle del Monte Barrochio, and devoted all his leisure to study. In his writings on statics he reproached the Schoolmen of the Xlllth Century, with good reason, for having made a first principle of gravitas secundum situm without having justified this action in any way. He wished to see substituted for this concept, the effect of the reaction of the support. " The mind cannot be at peace while the variation of gravity has not been attributed to some other cause than this. Indeed, it seems that [the variation secundum situm] is a symbol rather than a true reason. " The line CD resists a weight placed at D less than the line CL resists a weight at L. Thus, then, the same weight can be heavier or lighter in virtue of the effect of the posi tion it occupies ; not that by the very fact of this situation it really acquires a new gravity or that it loses its original gravity — rather it always keeps the same gravity in whatever place it may be ; but because it always weighs more or less on the circumfer- ence. " Guido Ubaldo confined himself to this qualitative statement, for he did not have at his disposal the law of the composition of forces. Nevertheless, he used the concept of moment to substantiate the condition for the equilibrium of a lever, by means of an argument whose form is directly inspired by Archimedes. He corrected certain errors in XVIth CENTURY 101 the Xlllth Century discussion of the stability of the balance, but made the mistake of using the same treatment when the verticals were assum ed parallel as he used when they were supposed to converge. Guido Ubaldo favoured Pappus' solution of the problem of the inclined plane — we have already seen the weakness and superficial character of this solution. Thus he was led to attribute a gravity to a moving body situated on a horizontal plane, contrary to the content of Xlllth Century statics. However, he in general preferred to consider virtual displacements than virtual velocities. He said that it is necessary to deploy a greater power in order to move a body than is necessary to maintain it in equilibrium, which shows that he did not understand the part played by the passive resistances. Guido Ubaldo took over Pappus' definition of the centre of gravity and supplemented it with the following commentary, which was to have a great influence on the authors of the XVIIth Century. " The rectilinear fall of bodies shows clearly that heavy bodies fall according to their centres of gravity. . . . Strictly speaking, a heavy body weighs through its centre of gravity. The very name centre of gravity seems to declare this truth. Clearly, all the force, all the gravity of the weight is massed and united at the centre of gravity ; it seems to run from all sides towards this point. Because of its gravity, indeed, the weight has a natural desire to pass through the centre of the Universe. But it is the centre of gravity that properly tends to the centre of the World. " Thus to Guido Ubaldo, just as much as to the writers of the XlVth Century, the concept of centre of gravity was a purely experimental one. It was not linked in any way with the parallelism of verticals. Guido Ubaldo's works, " sometimes erroneous, always mediocre, were often a regression from the ideas that had inspired the writings of Tartaglia and Cardan. " T However, this work is a milestone in the history of mechanics in that it had a direct stimulating influence on the great founders of mechanics, to whom it brought the content of the researches of Antiquity and the Middle Ages. Its value was at least that of a link with the past. 6. J.-B. VlLLALPAND (1552-1608) AND THE POLYGON OF SUSTENTATION. J.-B. Villalpand was born at Cordoba in 1552 and belonged to the Society of Jesuits. He became concerned with mechanics because of the diversion of an archeological mission to Jerusalem. He took it 1 DUHEM, 0. S., Vol. I, p. 226. 102 THE ORIGINS upon himself to refute certain of EzechiePs commentators, who had claimed that, because of its physical geography, Judea offered better possibilities for agriculture and construction than a plain of the same area would have done. This explains the title of Villalpand's book, Apparatus Urbis ac Templi Hierosolymitani, which was printed at Rome in 1603. In it Villalpand states, among others, the following proposition — " A heavy body that rests on the ground and covers a certain area remains in equilibrium when the vertical drawn through the centre of this area passes through the centre of gravity ; or, otherwise, when a vertical drawn through the edge of this area passes through the centre of gravity or leaves it on the same side as the area. But if it leaves the centre of gravity on the other side of the area, the heavy body will necessarily fall. " Here is his proof — " If the line FC, when produced, leaves the centre of gravity L of the body on the opposite side to the area BC upon which the heavy body rests, the body will necessarily fall. Indeed, the weight CLG is H Fig. 31 equal to the weight CLA. The weight CGH will be greater than the weight CHA. The heavier volume will drag the less heavy one along. . . , and the body will fall on the side of G. " It is quite probable that Villalpand, either directly or otherwise, borrowed this result, together with his later considerations on the walk- XVIth CENTURY 103 ing of living beings and the flight of birds, from da Vinci. However it may be, we are indebted to P. Mersenne for having made the preceding theorem on the polygon of sustentation classical. That tireless scholar was able to extract it from the religious exposition in which it had been lost, and to reproduce it in his collection Synopsis mathematica (Mecha- nicorum libri), published at Paris in 1626. f 7. J.-B. BENEDETTI (1530-1590). STATICS. FIGURE OF THE EARTH. DOCTRINE OF " IMPETUS. " From the start of his scientific career in 1553 Benedetti denied the truth of the following proposition of Aristotle, a proposition which had been adopted by Jordanus's School — Let two bodies, A and J5, be made of the same substance and let A have twice the volume of B. The velocity of fall for A is twice that for B. More generally, Benedetti rejected Aristotle's statics. " The laws of the lever, " he wrote, " do not depend in any way on the rapidity or on the extent of the motion. " This does not mean that he adopted Jordanus's doctrine, or in other words, that he substituted the concept of virtual work for that of virtual velocities. In fact, he reduced the whole of statics to the single rule of the lever and the concept of moment. " The ratio of the gravity of the weight placed at C to the gravity of the weight placed at F is equal to the ratio of BC to Bu. . „ . This will appear evident to us if we imagine a vertical thread jFu, and if we imagine that the weight at jF hangs from the end of the thread at M. It is clear that the weight hung in this way would produce the same effect if it were placed at F. " It seems that Benedetti had an inkling of the general utilisation of mo ments for measuring the effects of weights or of any motive powers whatever. To a certain extent then this criticism of Benedetti's was useful and constructive. On the other hand his rejection of the solution of the problem of the inclined plane, due to the unknown author of the Xlllth Century, and his repetition of Leonardo da Vinci's errors concerning the division of a weight between two convergent supports, was less fortunate. In the matter of the figure of the Earth and the separation between the continents and the oceans, Benedetti found his inspiration in Coper nicus. In 1579 he denied the truth of Albert of Saxony's opinions in the following terms. M Fig. 32 104 THE ORIGINS " We are certain that the spherical surface of the water is everywhere equidistant from the centre of the Universe, the point sought by all heavy bodies. Moreover, because of the numerous islands, because of the different countries which navigation has discovered in all regions, we can be sure and certain that the water and the earth comprise one globe, and that the geometrical centre of the Earth, together with the centre of its gravitation, is at the centre of the Universe. " We must add that Benedetti considered the copernican system to be a plausible one, though he did not accept it himself. It is said that Benedetti's works, united under the title Diversarum speculationum mathematicarum et physicarum and published in 1585, covered all the branches of mechanics. It remains to us to speak of Benedetti's important contribution to the doctrine of impetus. At the outset Benedetti maintained that a constant motive agency produced an accelerated motion. " In natural and rectilinear motion the impressio, the impetuositas recepta, increases continually, for the moving body contains in itself the motive cause, that is to say the pro pensity to take itself to the place to which it is assigned. Aristotle should not have said that a body moves more rapidly as it approaches its goal, but rather that a body moves more rapidly as it becomes further separated from its point of departure. For the impressio increases pro portionally as the natural motion is prolonged, the body continually receiving a new impetus. Indeed it contains in itself the cause of motion, which is the tendency to regain the natural place from which it has been torn by violence. " This qxiotation shows that even if Benedetti remain ed impregnated with Aristotle's ideas, he was not imprisoned by them. As we shall see, he was also able to amend Buridan's thesis. Benedetti believed that the entity which was conserved in motion was the impetus in a straight line. In his opinion a horizontal wheel, as exactly symmetrical as possible and resting on a single point, cannot have a perpetual motion of rotation. He gives four different reasons for this. The first is " that such a motion is not natural for the wheel. " The second is because of the friction at the support. The third, because of the resistance of the air. The fourth reason, which is the only truly important one, we shall quote from Benedetti's text.1 66 We consider each of the corporeal parts which moves on its own by means of the impetus which has been imparted to it. This part has a natural tendency to rectilinear motion, not to a curvilinear one. If a 1 Translated into French by DUHEM. XVIth CENTURY 105 particle chosen on the circumference of the aforesaid wheel was cut off from this body, there is no doubt that, at a certain time, this detached part would move in a straight line through the air. We can see this in the example of the slings which are used to throw stones. In these slings the impetus of motion which has been imparted to the projectile describes, by a kind of natural propensity, a rectilinear path. The stone which is thrown sets out on a rectilinear path along the line which is tangent to the circle which it describes at the outset, and which touches this circle at the point at which the stone was released, as it is reasonable to assume. " In short, Benedetti was the first to have clarified the idea that the impetus was conserved in a straight line. From this correct idea, however, he formed an incorrect conclusion. Thus he maintained that the motion of a wheel must slow down spontaneously, because its particles do not follow the rectilinear paths which they have an innate tendency to take. In fact, when the Schoolmen of the XlVth Century applied their doctrine of impetus indiscriminately to rectilinear and curvilinear motions, they confused two notions which a classical science should have distinguished ; the principle of inertia, or the conservation, in certain privileged connections, of the rectilinear uniform motion of an isolated material point ; and the principle of energy, which entails the conservation of the living force when these forces do no work. If he had the essential merit of having caught a glimpse of the principle of inertia, Benedetti, on the other hand, misunderstood part of the truth of Buridan's thesis. 8. GIORDANO BRUNO (1548-1600) AND THE COMPOSITION OF MOTION. Giordano Bruno is best known as a metaphysician. A remote disciple of Nicholas of Cues, he believed at the same time in the unity and the infinity of worlds. He illustrated this by means of a system of Monads which were at once material and spiritual, which were not born and did not perish, but combined with and separated from each other. He was burnt alive at Rome on February 17th, 1600 for his lampooning of the Papacy rather than, it seems, for his metaphysical ideas. Bruno, who taught at the College of France and accepted Coper nicus' system, was a determined adversary of aristotelian ideas. Thus he rebutted, in his Cena de le Ceneri (1584), Aristotle's objection to the motion of the Earth which had depended on the fact that a stone thrown vertically upwards fell again at its starting-point. This he 106 THE ORIGINS accomplished by an argument which was analogous to; but more precise than, that of Oresme. For this purpose, he visualises two men, one on the deck of a ship and the other on the bank, and each holding a stone in his hand. It is arranged that, at some instant, the hands are in sensibly the same position and that, then, the stones are allowed to fall simultaneously on the deck of the ship. The second man's stone will fall behind that of the first. For the stone belonging to the man on the ship " moves with the same motion as the ship. It has therefore a certain virtus impressa which the other does not possess . . . even though the stones have the same gravity ; though they traverse the same air ; though they start from points which are, as nearly as can be arranged, the same ; though they are subject to the same initial impact. " 9. BERNARDINO BALDI (1553-1617). STATICS AND GRAVITY " EX VIOLENTIA. " Bernardino Baldi was at once a theologian, archeologist, linguist and geographer. He was a familiar of Guido Ubaldo and, in mechanics, seems to have been influenced by Leonardo da Vinci and others. In 1582 he wrote Exercitationes in mechanica Aristotelis problemata which was not printed until 1621. Bernardino Baldi rejected the point of view of virtual velocities — that of Aristotle — in statics. u We cannot be sure that the admirable effect of a lever has as its cause the velocity which follows from the lengths of the arms. Indeed, what is the velocity of something that does not move ? Now the lever and the balance do not move when they are in equilibrium and nevertheless a small power can then support a large weight. It will be retorted that if a very great velocity is not apparent in very long arms, it will at least be potentially present. Now the force which maintains [the lever] maintains the action. " In a more positive way Baldi concerned himself with the equili brium of a tripod and, in this connection, gave the rule of the polygon of sustentation. He took the product of the weight of a body and the height of the centre of gravity as a measure of the effort necessary to overturn the body. He also discovered the correct law for the stability of the balance and made a study of the sensitivity of the balance. He accepted Leonardo's solution of the problem of the in clined plane without rectifying it. In dynamics Baldi distinguished between gravity by nature and gravity by violence, in which the influence of an external motive agency was concerned. In a projectile animated by a simple motion XVIth CENTURY 107 of translation, the centre of the natural gravity , J3, coincides with the centre of the gravity ex violently under the influence of an impulsion with direction BD. These two centres are " only distinct by rea soning and not in reality. " And Baldi adds — " Projectiles cease to move because the im pression whose nature and impetuosity governs them is in no way natural, but purely acci dental and violent. Now nothing which is violent is perpetual. ... As long as violence predominates, violent motion is entirely similar to natural motion — it is lower at the start ; later, by the very fact of the motion, it becomes more rapid ; then, as the impressed violence weakens bit by bit, it slows down ; finally the motion disappears at the same time as the impetus and the moving body comes to rest. " As Duhem has remarked,1 " this opinion is strange and not very logical. If one can assume that the natural gravity, which is a per manent motive agency, creates at each instant a new impetus, one cannot conclude from this that the artificial gravity, that is the impetus imparted by the motive agency, engenders in its turn an impetus of a second kind. " However strange it may be this thesis, handed on by Mersenne, was to be taken over by Roberval. Duhem has even followed its trail as far as Descartes. " 2 1 Studes sur Leonard de Fmci, Vol. I, p. 139. 2 Cf. a letter from DESCARTES to MERSENNE on April 26th, 1643, which discusses the question of whether a sword thrust is more effective if it is made with the point, the central part or that near the hilt of the sword. CHAPTER SEVEN XVIth CENTURY (Continued) XVIIth CENTURY TYCHO-BRAHE AND KEPLER 1. THE SYSTEM DUE TO TYCHO-BRAHE (1546-1601). While the students of mechanics of the Renaissance remained faithful to the Schoolmens' tradition and rehearsed their arguments without taking account of the observations that were available to them the astronomers were patiently accumulating a host of data that were to be seized by classical science for the formulation of the laws of dynamics. Tycho-Brahe occupies a prominent place among these observers because of the volume and the precision of his obser vations, which were the foundation upon which Kepler's laws were based. We must say a word here of his system of the World and of his ideas on dynamics. Tycho-Brahe rejected Ptolemy's system because of the complexity of its epicycles. He rejected Copernicus' system on the grounds that the comets observed in opposition to the Sun were not affected by the annual motion of the Earth. In his Astronomiae instauratae progymnasmata (1582) he wrote, " That heavy mass of the earth, so ill-disposed towards motion, cannot be displaced and agitated in this way without conflicting with the principles of physics. The authority of the Holy Scriptures opposes it. ... I have set out to examine seriously whether there is any hypothesis which is completely in accord with the phenomena and the mathematical principles without being repugnant to physics and without incurring the censures of theology. It has turned out as I had hoped. . . . " I believe, firmly and without reservation, that the motionless earth must be placed at the centre of the World, in accord with the XVIth AND XVIIth CENTURIES 109 feelings of ancient astronomers or physicists and the testimony of the Scriptures. I in no way assume, like Ptolemy and the Ancients, that the earth is the centre of the orbits of the secondary moving bodies. Rather I believe that the celestial motions are arranged in such a way that only the Moon and the Sun and the Eighth Sphere — the most distant of all — have the centres of their motions at the earth. The five other planets turn round the Sun as round their Chief and King, and the Sun is always at the centres of their spheres and is accompanied by them in its annual motion. Thus the Sun will be the law and the end of all these revolutions and, like Apollo among the Muses, it alone will determine all the celestial harmony of the motions which surround it. " Tycho-Brahe's initial faith in his system is embodied in the following formula. " Nova mundani systematis hypotyposis ab authors nuper adinventa qua turn vetus ilia Ptolemaica redundantia et inconcinnitas, turn etiam recens Coperniana in motu terrae physica absurditas excluduntur, omniaque apparentiis caelestibus aptissime correspondent. " However, this assurance is less obvious in a letter written to Roth- mann and dated February 21st, 1589, " If you prefer to make the earth and the seas, together with the moon, revolve ; if you wish that the earth, however ill-suited to motion and far below the stars it may be, behave like a star in the ethereal regions, you are certainly the master. . . . But are not earthly things being confused with celestial things ? is not the whole order of nature being turned upside- down? " Fundamentally it was religious prejudice that dictated the form of Tycho-Brahe's thesis, for he was too wideawake not to admit the super iority of the copernican system over that of Ptolemy. " I acknowledge that the revolution of the five planets, which the Ancients attributed to epicycles, are easily and at little cost explained by the simple motion of the Earth ; that the mathematicians have adopted many absurdities and contradictions which Copernicus set aside ; and that his system even agrees a little more accurately with celestial phenomena. " In order that the planets might turn about the Sun, Tycho-Brahe was obliged to assume that the rotation of the Sun round the Earth was due to an attraction that was different from that between the planets and the Sun. In dynamics he opposed the motion of the Earth with the objection that a stone dropped from the top of a tower fell at the bottom. Thus he did not appreciate the fallacies in this argument, which Oresme and Giordano Bruno had already indicated, though he was almost certainly unaware of their writings. 110 THE ORIGINS 2. KEPLER (1571-1631). THE GENERAL CHARACTER OF HIS CONTRI BUTION. It may seem strange that this review of the origin of mechanics should finish with Kepler's work. But if he is numbered among the classics for his three fundamental laws on the motion of the planets, his metaphysical tendencies and his ideas on dynamics place him in the scholastic tradition. Though a forerunner of Newton, his own inspiration were the writings of Nicholas of Cues. Kepler's character is most complex. A tireless calculator, he returned to the interpretation of observations without ever being discou raged, and rejected every law that allowed the slightest imprecision. With great wisdom he remarked that in the domain of Astronomy innovations were apt to lead to absurdities. By this he meant that the observations of the Ancients, however rough, should not be neglected. Though a disciple of Tycho-Brahe, he had no less respect for Ptolemy and was alive to the necessity of not adhering to the copernican system. His preconceived ideas, his errors, inconsistencies and illusions are not hidden from the reader. Occasionally his writings have the air of the confessional — thus he declares that his desire to succeed makes him blind, " cum essem caecus pro cupiditate. " l He compares scientific truth to a nymph who steals away after allowing herself to be seen, and quotes Virgil 2 in this connection. We see him sacrificing himself to metaphysics, seeking the reflection of preordained harmonies on every occasion, and even lending himself to astrology. Should we regard this as a fashion of the time, or as evidence of difficulties of quite another kind ? Indeed, the following declaration is attributed to Kepler — Astronomy would die of hunger if her daughter, Astrology, did not earn enough bread for two. . . . Kepler's first scientific work, Mysterium Cosmographicum, was published at Tubingen in 1596. The pythagorean influence which was to become apparent in all Kepler's thought emerges clearly from this youthful work. Thus he sought to incorporate the dimensions of the different planetary orbits into the copernican system by comparing them with the radii of spheres inscribed or circumscribed to five regular polyhedra. He assumed that the planets moved under the influence of an anima matrix localised in the Sun, whose action on the planets 1 Astronomia nova, p. 215. 2 " Malo me Galataea petit, lasciva puella Et fugit ad salices, et se cupit ante videri. " The question here is the discovery of the elliptic trajectories of the planets, Astro nomia nova, p. 283. XVIth AND XTIIth CENTURIES 111 was greater as they came nearer to the Sun. This property, confined to the plane of the ecliptic, is therefore inversely proportional to the distance. The same is true of the velocity produced, in accordance with the aristotelian dynamics to which Kepler remained faithful. At least Mysterium Cosmographicum had the merit of attracting the interest of Tycho-Brahe, who thereupon used Kepler in the analysis of planetary observations and, if the tradition is to be believed, charged Kepler with the task of preparing a new table of the planets. Kepler completed this task in 1627, with the publication of Tabulae Rudolphinae. 3. THE ORIGIN OF THE LAW OF AREAS. We shall now follow Kepler's fundamental work in theoretical astro nomy — Astronomia nova ocmoAoy^ro g,1 seu Physica Caelestis tradita commentariis de motibus stellae Martis ex observationibus G. V. Tychonis Brake, Prague, 1609. In this work Kepler seeks a theory of Mars which will take account of the observations in a precise way and which will be, at the same time, compatible with the systems of Ptolemy, Copernicus and Tycho-Brahe, The following is a very much abbreviated version of Kepler's demon stration of the law of areas in the case of eccentrics.2 In the figure a is the centre of the World^ that is, the Sun in the coper- nican system and the Earth in the other astronomical systems. The centre of the eccentric which the planet describes is at /?. (This term refers to the Earth in the copernican system and the Sun in the others.) The point y is the equant (punctum aequantis), or the point about which, according to Ptolemy's hypothesis, the " planet " appears to describe a circle with uniform velocity. Kepler draws this circle as a dotted line with the point y as centre and radius equal to that of the eccentric of centre /?. Further, like Ptolemy, Kepler assumes the bisection of the eccentricity, or that a/3 is equal to fiy. Starting from the aphelion (or apogee) 8 and the perihelion (or perigee) £, two very small arcs dip and sco are drawn in such a way that the points y>, a and o> are colinear. Then the lines yyj and yo> are drawn to cut the dotted circle in % and r respectively. " According to Ptolemy, since the entire circle vcp (equal to the eccen- 1 That is, " concerning the search for causes, " meaning at once dynamical and metaphysical causes. 2 Astronomia nova, Chapter XXXII, p. 165. 112 THE ORIGINS trie but with centre y (is a measure of the planet's period, then the arc v% will be a measure of the time the planet (mora) spends on the arc &p of the eccentric. " Kepler calls the arc 5y arcus itineris and the arc v%9 arcus temporis* The same is true for the arcs ea) and cpr* Having supposed the angle (5oc^ to be very small, Kepler writes 7^ the arc VY , ys the arc SGI (1) . —J — the arc the arc rep Fig. 34 XVIth AND XVIIth CENTURIES 113 Because of the bisection of the eccentricity, the length fid is the arithmetic mean of yd and a<5. But the arithmetic mean of two quan tities which are nearly equal to each other is just greater 1 than their geometric mean — this Kepler verifies by means of a numerical example. Then /9<5 (or yv) <x.d =^= and > — • yd fid From which follows through eq (1) ^-^ , ^ the arc vy , <x.d (2) 4 =£= and > — • the arc ipS fid In the same way, if /?£ is the arithmetic mean between and ye and oce, it is found that ys ^e =£= and < — • fie (or y(p) oce Hence, by (1), the arc sa> Be - ^ =£± and < — • the arc <pr as If then one considers, on the eccentric, two very small arcs dip and £O>, assumed to be equal to each other, each of them will be the mean pro portional between the arc v% — the time spent at the aphelion — and the <•— «v arc (pr — the time spent at the perihelion. Further, the ratio of the ^ ^ . //?e\2 arc v% to the arc (pr will be very nearly equal to ( — 1 • Or again, more clearly, if two very small and equal arcs dy and £O> are taken on the eccentric, the ratio of the times spent on the arcs will t> be the ratio of the arcs v% and 9?r, and will be equal to — , since, to the . , . a<5 fe0\* square in the eccentrics, — = — ) . ^ Now Kepler is in a position to state the law of areas for eccentrics. " Quanto longior est oc(5 quam oce, tanto diutius moratur Planeta in certo aliquo arcui excentrici apud 6, quam in aequali arcu excentrici apud s." That is, the greater a<5 is than as, the longer the planet will remain on a certain arc in the immediate neighbourhood of d than on an equal arc of the eccentric in the neighbourhood of £. 1 In the modern sense, " approximately equal to and greater than " or " =s= and >.'* 114 THE ORIGINS In the neighbourhoods of other points on the eccentric which are opposite to each other with respect to the centre of the world a, the behaviour of the planet is analogous, " quanto evidentior in demonstra- tione? tanto minor in effectu. " In fact, Kepler confined himself to the remark that the proportion of oc/^ to OLV is smaller, and that of a0 to oa is much smaller, than the proportion of oc<5 to oca. (See fig. 34.) Kepler translated this purely geometrical and kinematic analysis into dynamical terms in the very title of the chapter which we have analysed — Virtutem quam Planetam movet in circulum attenuari cum discessu afonte. (The strength — understood as the force — by means of which the Planet moves circularly falls off with the distance from the source [of motion].) This is evidence of the fact that Kepler remained faithful to aristotelian dynamics. Indeed, the force is measured in Kepler's mind by the inverse of a duration of sojourn on an arc, that is, by the velocity to which it corresponds. We see here the continuity of Kepler's views from his Mysterium Cosmographicum to his Astro- nomia nova. 4. ORIGIN OF THE LAW OF THE ELLIPTICITY OF PLANETARY TRAJECTORIES. Tycho-Brahe and Longomontanus had prepared a table of the oppo sitions of Mars since 1580. Tycho-Brahe, who had started Kepler on his study of the theory of Mars, himself represented the orbit of that planet by an eccentric whose geometrical centre did not bisect the eccen tricity. Now Kepler, either by tradition or because of his metaphysics, was attached to the hypothesis of the bisection of the eccentricity, which Ptolemy had put forward in connection with the major planets alone. He even went as far as to extend it to the Earth's orbit (in the context of Copernicus' system) and to that of the Sun (in the other systems). Kepler immediately started a methodical refinement of the values assigned to the radii of the Earth's orbit, which determined the scale of all the other interplanetary distances. He then turned his attention to Mars. Being unable to follow him through all the various detours that he made, we shall only record that he succeeded in accounting for all of twelve oppositions of Mars to within 2' of arc. This was accomplished by a painful method of trial and error in which four longitudes of Mars in opposition were used simultaneously. This necessitated, on Kepler's own confession,1 no 1 Astronomia nova, Chapter XVI, p. 95. XVIth AND XVIIth CENTURIES 115 less than seventy repetitions of the calculation. But the longitudes of Mars in position other than opposition invalidated the eccentric calcu lated in this way. Moreover, the eccentric did not satisfy the hypothesis of the bisection of the eccentricity — it transpired that the distances from the geometrical centre to the Sun and to the equant were not 7-232 equal, but were in the ratio — ^ 11-332 Returning to the hypothesis of the bisection of the eccentricity for the orbit of Mars, and relying on the observation of the opposition of Mars in 1613, Kepler found an error of about 8' in the annual parallax of the planet. Fortunately for theoretical astronomy and for the deve lopment of newtonian mechanics, Kepler refused to neglect such a disparity between calculation and observation. He proceeded to evaluate the distances from Mars to the Sun in terms of the distances from the Earth to the Sun. The accompanying diagram shows how a knowledge of the longitudes of Mars and of the Earth, together with a knowledge of the two radii (SB and SC) of the Earth's orbit, allow the distance from the Sun to Mars (SM) to be de termined. In the diagram the circle with centre 0 represents the Earth's Fi£* 35 orbit, S the Sun and M, Mars, while B and C are two positions of the Earth for the same position of Mars. In particular, Kepler proceeded in this way for the distances from Mars to the Sun in the neighbourhood of the aphelion and of the perihelion, and thus obtained the eccentricity of the planet. Kepler compared these observations with a circular eccentric satis fying the principle of the bisection of the eccentricity. He established a systematic failure of distances with respect to the circumference of the circle, " Itaque plane hoc est ; orbita planetae non est circulus, $ed ingre- diens at latera utraque paulatim, iterumque ad circuit amplitudinem in perigeo exiens, cujusmodi figuram itineris ovalem appellitant. " x Only observation could make Kepler give up the hypothesis of the circle, which was based on the authority of the ancients and, for the rest, agreed with his own metaphysics. At first Kepler was reluctant to make an ellipse of this oval orbit, 1 Astronomia nova. Chapter XLIV, p. 213. 116 THE ORIGINS though he did investigate whether a particular ellipse that he had chosen could reconcile the data, only to discover that this was not so. 1 Finally, however, after many unsuccessful attempts, he wrote, " Inter circulum vero et ellipsin, nihil mediat nisi ellipsis alia " (between a circle and an ellipse there can be nothing but a second ellipse). And he con cludes that " Ergo ellipsis est Planet ae iter. " 2 5. KEPLER'S THIRD LAW. The extremely important positive success of the theory of Mars did not turn Kepler's interest away from astrology and metaphysics. Thus more than ten years elapsed before Harmonices Mundi was published at Linz in 1619. Of the first five books in this work only the last makes mention of astronomy, and even this is confused with strange meta physical conceptions. For example, we see him develop an analogy between the angular velocities of the planets about the Sun and the frequencies of musical notes, and expressing the oscillation of these angular velocities during the course of a revolution by means of a musical notation. The question is really one of pythagorean harmony, with the reservation that it remains abstract and that Kepler did not pretend that it was perceptible by our senses. In following the Astronomia nova we have seen that Kepler carne across the law of areas before that of ellipticity. It has however become customary to reverse the order in which these two laws are presented, and to forget that Kepler, without justification, extended the law of areas to elliptical trajectories allthough he had only established it for eccentrics. Kepler's third law is stated in Chapter III of Book V of Harmonices MundL3 He recalls the fruitless efforts that he had made, since the beginning of his scientific career, to establish a connection between the periods of the planets and the dimensions of their orbits. Not until March 8th, 1618, did he come across the characteristic ratio in this law — a gross error of calculation made him reject it at first. Finally he persuaded himself of its correctness — " Res est certissima exactissimaque, quod proportio quae est inter binorum quorumcunque Planetarum tempora periodica, sit praecise sesquialtera proportionibus mediarum distantiarum^ id est Orbium ipsorum. " (One thing is absolutely certain and correct, 3 that the ratio between the periods of any two planets is, to the power ~, £ 1 Astronomia nova, Chapter XLV. 2 Ibid., Chapter LV, p. 285. 8 P. 189. XVIth AND XVIIth CENTURIES 117 exactly that of their mean distances, that is, of their orbits.) This quite empirical result may be written 6. KEPLER AND THE CONCEPT OF INERTIA. Kepler had the merit of having emphasised the concept of inertia more completely than his predecessors had done — indeed it is sometimes maintained that he actually formulated the principle of inertia. This is not true in the sense that Kepler's concept of inertia remained linked with Aristotle's mechanics and with Buridan's doctrine as modified by the German School of the XVth Century. " The proper characteristic of material which forms the greatest part of the Earth is the inertia. Motion is repugnant to it, and more so as a great quantity of material is confined in a smaller volume. " 1 Kepler adds — " This material inertia of a terrestrial body, this density of the same body, constitute exactly the subject on which the impetus of rotational motion is impressed. It is impressed there exactly as in a top which turns because of violence. The heavier the material of the top is, the better it assimilates the motion impressed by the external force and the more lasting this motion is. " 2 Kepler's dynamics follows directly from the ideas of Nicholas of Cues, whom he called " divinus mihi Cusanus. " He took up the example of the toy top which Nicholas of Cues had given, and applied the doctrine of impetus impressus to celestial bodies. " Could not God have produced [such an impetus impressus] in the Earth, as from the exterior, at the beginning of time ? It is this im pression which has produced all the past rotations of the Earth and which maintains them even now, though their number already exceeds two millions. Indeed, this impression keeps all its vigour because the rotation of the Earth is not hindered by impact or by any external roughness ; or by the ethereal fluid, which is devoid of density. No more is it hindered by any weight, or by any internal gravity. As for the inertia of the material, that is the very subject which receives the impetus and conserves it as long as the motion continues. " 3 1 Opera omnia, Vol. VI, p. 174. 2 Ibid., p. 175. 3 Ibid., p. 176. 118 THE ORIGINS Kepler believed that the material of the Earth was separated into circular fibres whose centres were aligned with the axis of rotation. " This arrangement of the Earth into circular fibres predisposes it to the motion that it receives. All the same, it appears that these fibres are the instruments of the motive cause rather than the motive cause itself. " i The impetus communicated to the Earth by the Creator becomes a soul. " It is a soul of a strange kind. It confers on the Earth neither growth nor discursive reason (sic) — it merely moves it. But, better than a simple corporeal faculty, this motive soul assures the perfect regularity of diurnal motion. This motion, indeed, is no longer a vio lent motion, in any sense, for the Earth. What is there, indeed, more natural to a material than its form, to a body than its faculty or soul ? " 2 7. KEPLER AND THE DOCTRINE OF ATTRACTION. Following the example of Copernicus,3 Kepler showed himself to be a Pythagorean in the matter of gravitation. He therefore denied the thesis that Albert of Saxony had made classical since the XlVth Century. " The doctrine of gravitation is erroneous. A single mathematical point, whether it be the centre of the World or any other point, cannot effectively move heavy bodies, nor be the object towards which they tend. Therefore let them, the Physicists, prove that such a force can belong to a point, which is not a body and which is only conceived in an entirely relative way. " It is impossible that the [substantial] force of a stone, which sets the body in motion of itself, should seek a mathematical point, the centre of the World without regard to the body in which that point may be situated. Therefore let them, the Physicists, establish that natural things have sympathy for that which does not exist. " 4 And Kepler expounds " the true doctrine of gravity. " " Gravity is a mutual affection between parent bodies (Gravitas est affectio corporea, mutua inter cognata corpora) which tends to unite them and join them together. The magnetic faculty is a property of the same kind. It is the Earth which attracts the stone, even though it might not tend towards the Earth. In the same way, if we place the centre of the Earth at the centre of the World, it is not towards the centre of the World that bodies are carried, but rather towards the 1 Opera omnia, Vol. VI, p. 178. 2 Ibid., p. 179. 8 See above, p. 85. 4 Astronomia nova, Introductio, para. VIII. XVIth AND XVIIth CENTURIES 119 centre of the body around which they belong, that is to say, the Earth. Also, the heavy bodies will be carried towards whatever place the Earth is carried to, because of the faculty which animates it. " If the Earth was not round, heavy bodies would not move directly towards the centre from all directions. But according to whether they come from one place or another, they will be carried to different points. " If, in a certain position in the World, two stones are placed near each other and outside the sphere of attraction of all other bodies which could attract them, these stones, like two magnets, will tend to unite in an intermediate position and the distances they will travel in order to unite will be in inverse ratio to their masses. " 1 1 Ibid. PART TWO THE FORMATION OF CLASSICAL MECHANICS CHAPTER ONE STEVIN'S STATICS SOLOMON OF CAUX 1. THE STATICS OF STEVIN (1548-1620). Stevin's first work on statics was published in Flemish at Leyden in 1586, under the title De Beghinselen der Weegconst* A more complete version appeared in 1605. Finally, in 1608, Stevin united these works under the title of Hypomnemata Mathematica. This work was trans lated into French as early as 1634. Stevin's statics is developed geometrically in a manner similar to that used by Archimedes. In it, the author systematically neglects " the motions of machines, formed of wood or iron, in which certain parts are lubricated with oil or lard, others are swollen by the humidity of the air or corroded with rust, in which these varied circumstances and also many others some times facilitate the motion, sometimes hinder it. " Moreover, Stevin refuses to consider the excess of motive power which motion demands, " for the obstacles to motion have no certain and unique relation with the object moved. " Still more rigorously, Stevin rejected the consideration of arcs of a circle described by the ends of the arms in the problem of the equilibrium of a lever. And he justified this by means of a syllogism. " Something which does not move does not describe a circle. Two weights in equilibrium do not move. Therefore two weights in equi librium do not describe circles. " We see that Stevin eschewed the point of view of virtual velocities in order to romp in the field of pure statics. At least he imposed this restriction on the form of his writing. He was not, however, to main tain it exclusively, as we shall show. On the subject of the lever, Stevin added some further refinements to Archimedes' demonstrations which we shall pass over. 124 THE FORMATION OF CLASSICAL MECHANICS He solved the problem of the equilibrium of a heavy body on an inclined plane by a method that was completely original and which was based on the impossibility of perpetual motion. This is his demonstration, taken from the French edition of 1634. " Given. — Let ABC be a triangle whose plane is perpendicular to the horizon and whose base AC is parallel to the horizon. Let a weight D be placed on the side AB, which is to be twice J3C, and a weight JS, equal to D, be placed on the side BC. " The Requirement. — It is necessary to show that the power (or capacity of exerting power) of the weight E is to that of the weight D as AB is to BC, that is, as 2 is to 1. " Construction. — Round the triangle let there be arranged a system of fourteen spheres equal in weight, size, and equidistant from each other at the points D, E, F P, Q, B, and threaded on a cord passing through their centres in such a way that there are two spheres on BC and four on BA .... Let S, T, V be three fixed points on which the cord can run freely without being caught. " Demonstration. — If the power of the weights D, JR, Q, P were not equal to the power of the weights E, F, one of the sides would be heavier than the other. Suppose then that the four D, jR, Q, P are heavier than the two E9 L. Now the four 0, N, M, F are equal to the four G, H, /, K. Now the side with eight spheres D, R, Q, P, 0, JV, M, L STEVIN AND CAUX 125 will be heavier than that of the six spheres JS, .F, G, H, I, K and, since the heavier part will dominate the lighter, the eight spheres will fall and the six will rise. Thus D will come to where 0 is at present and the others will do the same. That is, that £, F, G, H come to the positions where P, Q, .R, D are now and JT, K to where jE, jF are. How ever the effect of the spheres will have the same disposition as pre viously and for the same reason the eight spheres will weigh more and, when they fall, will make eight others come in their place. Thus this motion will have no end, which is absurd. The demonstration will be the same in the opposite case. Therefore the part jD, R . . . . L of the ring will be in equilibrium with the part J5, JF, .... 1C. If there be taken away from both sides the heavinesses which are equal and similarily arranged, the four spheres 0, JV, M, L on the one hand and the four G, fl, I, K on the other, the four D, It, (), P which will be left will be in equilibrium with the two .E, F. Hence E will have a power twice that of D. Therefore the power of E is to that of D as the side BA9 let it be 2, is to the side BC, let it be 1. " Stevin had read Cardan and referred to his Opus novum. He could not have been ignorant of De Subtilitate in which Cardan, after Leonardo da Vinci, held that perpetual motion was impossible. Stevin was legitimately proud of his demonstration. He reproduced the associated diagram in the frontispiece of Hypomnemata Mathema- tica with the legend " Wonder en is gheen wonder " (The magic is not Fig. 37 magical), no doubt intending to indicate that he had logically explained a fact instead of invoking magic as the Greeks had done, before Archi medes, in connection with levers. From this theorem on the inclined plane Stevin deduced the value of the weight E which could support the column D on an inclined 126 THE FORMATION OF CLASSICAL MECHANICS plane by means of a thread parallel to the plane and starting from the centre of gravity of the column. The result was given by D ~E AB BC He then studied a series of more complicated examples, like the following one in which the direct elevation, M, in equilibrium with the column D is compared with an oblique elevation, such as E, which is able to hold the column on the inclined plane. In these circumstances M is equal to D, and the preceding result v w D AB is applicable : -= = ^7= • E CB BC DI AT N°W ^ m * M DL Therefore- = — Fig. 38 Thus Stevin, after many false starts, arrived at an enunciation and even a verification of the rule of the parallelogram of forces for the particular instance in which the two forces are at right angles. Let a column of centre of gravity C be hung from the points D and E by means of two strings CD, CE. Complete the parallelogram CHIK whose diagonal CI is vertical. " The direct elevation is to the oblique elevation as CI is to CH. But the direct elevation CI is equal to the weight of the column. Therefore the weight of the whole column to the weight which occurs at D is as CI is to CjfiT. In the same way, the weight which occurs at E will be found by producing the line IK from I parallel to DC to meet CJE. STEVIN AND CAUX 127 In words, the weight of the column which occurs at E will be as the straight elevation CI is to the oblique elevation CK. " Fig. 39 From these considerations Stevin deduced the tension of the threads of a funicular polygon and thus became the originator of graphical statics. 2. STEVIN AND THE PRINCIPLE OF VIRTUAL WORK. Returning to the point of view which had turned statics into a purely deductive science, and starting only from the assumption of the impossibility of perpetual motion, Stevin quite clearly stated the principle of virtual work. This occurs in volume IV of his Hypomnemata in connection with Stevin's work on the equilibrium of systems of pulleys. " The distance travelled by the force acting is to the distance tra velled by the resistance as the power of the resistance is to that of the force acting. (Ut spatium agentis ad spatium patientis, sit potentia patientis ad potentiam agentis) . " 128 THE FORMATION OF CLASSICAL MECHANICS 3. STEVIN'S HYDROSTATICS. Stevin's contribution to hydrostatics is quite remarkable. He clearly stated the principle of solidification according to which a solid body of any shape and of the same density as a given fluid can remain in it at equilibrium whatever its position may be, and without the pressures in the rest of the fluid being modified. He used this principle to determine the pressure on each element of the base by solidifying all the liquid except that in a narrow channel abutting on this element, and verified that this pressure was independent of the shape of the receptacle and depended only on the weight of the column of liquid which filled the channel. This led him to state the hydrostatic paradox — that a fluid, by means of its pressure, can exert a total effort on the bottom of a vessel which can be considerably greater than the total weight of the fluid. He also determined the resultant of the pressures on an inclined plane boundary wall by dividing this surface into hori zontal slices and passing to the limit by increasing the number of slices indefinitely. Finally, he related Archimedes' principle to the impossibility of perpetual motion. Thus he was guided by the same idea as in the problem of the inclined plane. 4. SOLOMON OF CAUX (1576-1630) AND THE CONCEPT OF WORK. Solomon of Caux was a practical Norman who was concerned with the construction of hydraulic screws. In 1615, after having read Cardan, he published at Frankfurt a work entitled The reasons of moving forces together with various machines, as much useful as plea sant , to which are added some designs of grottoes and fountains. It is to this author that we owe the term work in the sense that it is used in mechanics now. CHAPTER TWO GALILEO AND TORRICELLI 1. GALILEO'S STATICS. Galileo (1564-1642) started his scientific career in the way that was customary in his time, by annotating Aristotle's De Caelo. His manuscript remained unpublished until 1888 and is a typical scholastic document, even though it refers to certain moderns like Cardan and Scaliger. Nevertheless, holding a chair of mathematics at the University of Pisa at the early age of twenty- five, Galileo was not long in causing a scandal by publicly experimenting on the fall of heavy bodies, by attacking his elders, and by offending a natural protector like John de Medici by wounding his pride in his inventions. However, his fierce intellectual independence, which, in its turn, was to earn him many rebuffs, developed very rapidly. He did not long remain a slave to the scholastic discipline. We shall not concern ourselves here with Galileo's biography, which belongs to the general history of science, but shall make an analysis of his contribution to statics and dynamics. First we shall follow the Mechanics of Galileo in the French version which Mersenne published at Paris in 1634. Chronologically this work lies between the manuscript of Galileo's lectures at Padua in 1594 and the treatise Delia Scienza meccanica which was printed at Ravenna in 1649, seven years after its author's death. At the beginning of the Mechanics Galileo emphasises " that ma chines are useful for manoeuvring great loads without dividing them, because often there is much time and little force . . . but he who would shorten the time and use only a little force will deceive himself. " That is, Galileo considers the product of the force and the velocity in conformity with Aristotle's thesis. To Galileo the heaviness of a body was a " natural inclination of the body to take itself to the centre of the Earth. " D 130 THE FORMATION OF CLASSICAL MECHANICS The moment was the inclination of the same body considered in the situation which it occupied on the arm of a lever or a balance. It is " made up of the absolute heaviness of the body and its separation from the centre of the balance, and corresponds to the Greek QOTZIJ. " The existence of a centre of heaviness (centre of gravity) of a body was just as much an experimental fact to Galileo as it was to the School men. " Each body principally weighs through the centre in which it masses and unites all its impetuosity and weight. " In turn, Galileo studied the lever, the steelyard, the lathe, the fly wheel, the crane, the winch, the pulley and the screw. The discussion of the screw entailed a study of the inclined plane and, in this connection, Galileo /• was able to do better than his prede cessors had done. He envisages a perfectly round and polished ball to be placed on a per fectly smooth surface. On the hori zontal plane AB " the ball is indifferent to motion and rest, so that the wind or the smallest force can move it. " But w — — — ^ u a greater lorce is necessary in order to Flg> 40 lift the ball on the inclined planes AC, AD, AE and finally, " it will only be possible to lift the ball on the perpendicular plane with a force equal to its whole weight. " Galileo proceeds by considering two equal weight, A and C, in equilibrium on the lever ABC. In this way he is able to amend Pappus's demonstration, which he cites in this connection. If the arm BC falls to JBF, the moment of the weight F becomes less than the moment BK of the equal weight A in the ratio p~^* " When the weight is at F it is partly maintained by the circular plane CI and its slope — or the tendency which it has to the centre of the earth is diminished by the extent that BC exceeds BK. So that it is supported by this plane to the same extent as if it had been supported by the tangent GFH? more especially as the slope of the circumference at the point F only differs from the slope of the tangent GFH by the insensible angle of contact. " By means of this remarkable artifice Galileo reduces the effect of the weight F on the inclined plane GFH to the effect of the same weight suspended as if from the arm of the lever BF. And he concludes that " the ratio of the total and absolute moment of the moving body, GALILEO AND TORRICELLI 131 in the perpendicular to the horizon, to the moment which it has on the inclined plane HF is the same as the ratio of FH to FK. BF . ,FH For 1S 6q ~FK.' Galileo also solves the problem of the inclined plane by appealing, this time, to the concept of virtual work. "imagine that in the triangle ABC the line AB represents the horizontaf plane, the line AC the inclined plane whose height mil be measured by CB. On the plane AC is placed a body E attached to the string EDF. At F the string carries a weight, or a force, which is related to the weight E in the ratio of the line BC to the line CA. If the weight F starts to fall, drawing the body E along the inclined plane, the body E will travel a path in the direction of AC which is equal to that which the heavy body F describes in its fall. But the Fig. 42 body escres n . following observations are necessary. It is true that the body E have travelled all the line AC in the same time that the weight F have SL to fall an equal distance. But during this time, the body E will not have been separated from the common centre of heavy 132 THE FORMATION OF CLASSICAL MECHANICS things by a distance greater than the vertical J5C, while the weight F, falling vertically, will have fallen a distance equal to the whole line AC. Now the bodies only resist an oblique motion to the extent to which they are taken away from the centre of the earth We can legitima tely say that the path of the force F keeps the same ratio to the path of the force E as the ratio of the length AC to the length CJ5, and is therefore equal to the ratio of the weight E to the weight F. " 2. GALILEO AND THE FALL OF BODIES. Thanks to a letter which Galileo wrote to Paolo Sarpi, dated October 16th, 1604, x we know that as early as this Galileo believed in the now classical law of distances 5 = constant X t2. " The distances gone through in natural motion are in square ratio to the times of fall. Consequently the distances travelled in equal times are related to each other like the consecutive odd numbers starting from unity. " Nevertheless, at first Galileo associated this law of distances with an incorrect law of velocities, namely v = k - 5. "We recall that as early as the XlVth Century Albert of Saxony had hesitated between this law and the correct one, v=k*t. To Galileo, the law v = k • s was explained in the following way. " A body which moves naturally increases in velocity to the extent that it is separated from the source of its motion. " The arguments by which Galileo sought to verify these two laws simultaneously are \k " rather odd. Certainly they are incorrect, but they show us the development of his thought and deserve to be quoted as showing what detours he made before he became eman cipated from them. " If the heavy body starts from the point A and falls along the line AB, I suppose that the degree of velocity at the point D exceeds the degree of velocity at the point C in the ratio of DA to CA ; that in the same way, the degree of velocity at E is to the degree of velocity at D as EA is to DA. Thus, at every point of AB the body will have a velocity proportional to the 1 The Works of Galileo, Italian National Edition, Vol. X, p. 115. f B 7 43 GALILEO AND TORRICELLI 133 distance from this same point to the origin A. This principle appears to me to be very natural. It corresponds to all the observations that we make of machines whose purpose is hitting. Given this principle, I will demonstrate the rest. " Let the line AK make any angle with the line AF and, through the points C, D, E, F let the parallels CG, DH, EI9 FK be drawn. Since the lines FK, El, DH, CG have the same relation to each other as the lines FA, EA, DA, CA, the velocities at the points F, E, D, C are therefore related to each other like the lines FK, El, DH, CG. There fore the degrees of velocity at all the points of the line AF are constantly increasing according to the increasing of the parallels drawn from these same points. " Moreover, since the velocity with the moving body goes from A to D is made up of all the degrees of velocity acquired at all the points of the line AD, and since the velocity with which it has travelled the line AC is made up of all the degrees of velocity acquired at all the points of the line AC, the ratio of the velocity with which it has travelled the line AD to the velocity with which it has travelled the line AC is that between all the parallels drawn from all the points of the line AD to the line AH. This [ratio] is that of the triangle ADH to the triangle ACG, that is, the ratio of the square of AD to the square of AC. Therefore the relation of the velocity with which the moving body has travelled the line AD to the velocity with which it has run through the line AC is the square of the ratio of DA to CA. " But the ratio of a velocity to a velocity is the inverse of the ratio of the corresponding times, for the time decreases when the velocity increases. The ratio of the duration of motion along AD to the dur ation of motion along AC is therefore the square root of the ratio of the distance AD to the distance AC. Therefore the distances from the starting point are as the squares of the times. However, the distances travelled in equal times are to each other as the consecutive odd numbers starting from unity. This is in accord with what I have always said and with observations made. All the verities are thus in accord. " l Briefly, from the inexact hypothesis that v = k • s, Galileo obtains ^ l . v(D) DA the relation -77^ = -— • v(C) CA Then by a consideration of a series of parallels erected from each f ATX 1. J J • 1 T. , - VAD DA* point ol AM he deduces, incorrectly, the relation v(AC) \CA where v(AD) and v(AC) are the mean velocities on AD and AC. 1 Complete Works of Galileo, Italian Edition, Florence 1908, Vol. VIII, p. 373. 134 THE FORMATION OF CLASSICAL MECHANICS From the last relation, and again incorrectly, he concludes that the ratio t(AC}~ V CA and thus arrives at the correct law 5 = constant X t2. We shall now follow the treatise Discorsi e dimostrazioni matema- tiche intorno a due nuove scienze attenanti alia Meccanica ed i movimenti localL The first edition of this work appeared in 1638 and was later supplemented by the author, although these additions only appear in an edition that was printed at Bologna in 1655. In these Discorsi three characters, Salviati (Galileo), Sagredo (a Venetian senator and friend of Galileo) and Simplicio (who represents Scholasticism) discuss the work. This dialogue form has the obvious inconvenience of making the book difficult to read, but has the in estimable advantage of allowing the author to show the development of his thought. The Text declares that the fall of bodies is uniformly accelerated, or that the " increase of the velocity is like that of the time. " Starting from rest, the moving body receives equal degrees of velocity. This the Text assumes a priori. " Why indeed not believe that the increases in velocity follow the most simple and banal law ? " Refusing to lose himself in the discussions which had occupied the Schoolmen, Salviati brushes aside all argument on the cause of the fall of bodies. He recalls that he had, for some time, believed that the velocity could increase as the distance did. As we have just seen, this was his opinion in 1604. He now rejects this belief. " If the velocities are proportional to the distances travelled, the distances will be travelled in equal times. Therefore, if the velocities with which the body travelled the 4 cubits were double those with which it travelled the first two cubits (as the distances are doubled) the durations of travel will be equal. But the same moving body can only travel the 4 or the 2 cubits in the same time if this motion is instan taneous. Now it is apparent that the motion of a heavy body lasts a certain time, and that it travels the first two cubits in less time than the four. Therefore it is not true that its velocity increases as the distance. " GALILEO AND TORRICELLI 135 We note here, with Jouguet,1 that this argument of Galileo is not quite correct. The law v = k-s immediately leads to s = s0exp(kt). In order that there should be motion it is necessary that, contrary to the hypothesis, s0 should be different from zero when t = 0. Other wise it is necessary to assume that in the first instant the body travels the distance s0 instantaneously. Given this, Salviati makes the following postulate. " I assume that the degrees of velocity acquired by the same moving body on differently inclined planes are equal whenever the heights of the planes are equal. " The moving body is assumed to be perfectly smooth and the planes to be perfectly polished. In order to substantiate this principle Salviati starts from the follow ing postulate, in which Galileo's physical intuition is very apparent. " Imagine that this sheet of paper is a vertical wall, that a nail is fixed in it and that a ball of lead weighing an. ounce or two is hung from the nail by a thread AB. The thread is to be two or three cubits long, perpendicular to the horizon and at a distance of about two fingers from the wall. Draw a horizontal CD on the wall to cut the thread AB squarely. Draw aside the thread AB and the ball into the position AC. Then release the ball. We will see this descend, describ ing the arc CB, and pass the extremity B in such a way that it will go up again, along J3D, almost to the line CD which has been drawn. Each time there will be a small deficiency, and this circumstance is precisely due to the resistance of the air and of the thread. From this we can conclude, in all truth, that the impeto at the point B which is 1 JOUGUET, L. M., Vol. I, p. 96. 136 THE FORMATION OF CLASSICAL MECHANICS acquired by the ball in its descent of the arc CB is such that it suffices to make it remount the identical arc BD to the same height. When this observation has been repeated again and again, fix in the wall a nail which projects about five or six fingers, exactly opposite the vertical AB — for example, at E or at F. The ball will describe the arc CB, the thread turning as before. When the ball comes to I?, the thread will tangle in the nail E and the ball will be obliged to travel the circum ference BG which has E as centre. Then we see that this can produce, at the extremity JB, the very impeto that can make the moving body rise again along the arc BD until it almost reaches the horizontal CD. Now, gentlemen, you will see with pleasure that the ball attains the horizontal at the point G. The same thing would happen if the nail were fixed lower, at F for example. The ball will describe the arc B I and will always finish its ascent on the line CD. And, if the nail were too low for the ball to attain the height CD (this would happen if the nail were nearer jB than CD) the thread would wrap itself round the nail. This observation prevents one from doubting the truth of the principle that has been supposed. Since the two arcs CJ?, DB are equal and similarly placed, the momenta acquired at jB along CB suffices to make the same body rise again along BD. Therefore the momenta acquired along D.B is equal to that which would make the same moving body rise again, along the same arc, from JB to D. So that in general, the momenta acquired along any arc is equal to that which can make the same body rebound along the same arc. But all the mamenti which make the body rebound along the arcs J5D, B G, BI are equal, since they are produced from the momento acquired in the descent CB, as obser vation shows. Therefore all the mamenti acquired in descending the arcs DB, GJB, IB are equal. " Salviati goes on to consider motions along variously inclined planes. " We cannot show with the same clarity that the same thing will happen when a perfect ball falls along inclined planes that are drawn along the chords of these same arcs. On the contrary, since the planes form an angle at the point J59 it is plausible that the ball, having descend ed along the chord CjB and meeting an obstacle at the bottom of the planes which mount along the chords JBD, BG, J5J, will lose a part of its impeto in rebounding, and will not be able to ascend again to the height of the line CD. But since the obstacle raised in this way prevents the observation, it seems to me that the mind will go on believing that the impeto (which contains, indeed, the force of the whole fall) will be able to make the body go up again to the same height. Therefore take this assertion as a postulate for the moment — its absolute truth will be established later when we shall see that the conclusions depending GALILEO AND TORRICELLI 137 Fig. 45 on this hypothesis are, in detail, in conformity with observation. " Galileo then established the now classical laws of falling bodies. In particular, we shall describe how, going back on his opinion of 1604, he established the law of velocities. 66 Since, in a accelerated motion, the velocity is continuously augment ed, the degrees of the velocity cannot be divided into any determinate number. For since the velocity changes from mo ment to moment and increases continuously, they are of infinite number. However, we can represent our intention better by constructing a triangle ABC, taking as many equal parts AD, DE, EF, FG as we please on the side AC, and in drawing straight lines parallel to the base BC through the points D, E, F, G. Then, if the parts marked on the line AC are equal times, we assume that the parallels drawn through the points D, J5, F represent the degrees of the accelerated velocity, degrees which increase equally in equal times. . . . " But because the acceleration is continuous from moment to moment and not of a discontinuous kind of this or that duration . . . before the moving body attains the degree of velocity DH that is acquired in the time AD, it has passed through an infinity of smaller and smaller degrees gained in the infinite number of instants that the time AD contains and which correspond to the infinity of points that lie on the line DA. However, in order to represent the infinity of degrees of velocity that precede the degree DJEf, it is necessary to imagine an infinity of lines, always smaller and smaller, which should be drawn from the various of the infinite number of points of the line DA. In ultimo, this infinity of lines will represent the surface of the triangle AHD. 64 Complete the whole parallelogram AMBC and produce as far as the side JBM, not only the parallels which have been drawn in the triangle, but also the infinite number of parallels that was imagined to start from all the points of the side AD. The line BC, which is the longest parallel drawn in the triangle, represents the highest degree of the velocity acquired by the moving body in its accelerated motion. The total surface of the triangle is the mass and the total of all the velocity with which the body has travelled such a distance in the time AC. In the same way the parallelogram will be the mass and the union of degrees of velocity each of which is equal to the maximum degree BC. This latter mass of velocities will be twice the mass of the increasing velocities of the triangle, because the parallelogram is twice the triangle. Conse- 138 THE FORMATION OF CLASSICAL MECHANICS quently, if a moving body takes degrees of an accelerated velocity, in falling, which conform to the triangle ABC, and if it passes through such a distance in such a time, it will, when moving uniformly, travel twice the distance that it has travelled in the accelerated motion. " By an analogous argument whose detailed reproduction would serve no useful purpose, Galileo arrived at the following theorem. " If a body starts from rest and moves with uniformly accelerated motion, the time that it takes to travel a certain distance is equal to the time that the same body would take to travel the same distance with a uniform motion whose degree of velocity was half of the greatest and final degree of the velocity of the uniformly accelerated motion. " We know that the Schoolmen, thanks to the efforts of Oresme, Heytesbury and Soto, had already obtained this fundamental result. But Galileo did not confine himself to the a priori assertion that the fall of bodies was uniformly accelerated. He submitted the fall of a body on an inclined plane to an experiment which was, for the time, per formed in a scrupulous manner and repeated a hundred times. We shall quote this essential passage of the Discorsi, noting its very marked difference from the tendencies of purely rationalist Scholasticism. " In the thickness of a ruler, that is, of a strip of wood about twelve cubits long, half a cubit wide and three fingers thick, a channel, a little wider than one finger, was hollowed out. It was made quite straight and, in order that it should be polished and quite smooth, the inside was covered with a sheet of parchment as glazed as possible. A short ball of bronze that was very hard, quite round and well- polished, was allowed to move down the channel. The ruler, made as we have des cribed, had one of its ends lifted to some height — of about one or two cubits — above the horizontal plane. As I have said, the ball was allowed to fall in the channel and the duration of its whole journey was observed in the way that I have explained. The same trial was repeated many times in order to be quite sure of the length of this time. In this repetition, no difference greater than a tenth of a pulse was ever found. When this observation had been repeated and established with precision, we made the ball fall through only a quarter of the length of the channel, and found that the measured duration of fall was always equal to half of the other. . . . " When this observation had been repeated a hundred times, the distances travelled were always found to be in the ratio of the squares of the times, and this was true whatever the inclination of the plane, or that of the channel in which the ball fell, was made to be. We also observed that the durations of fall on differently inclined planes were in the proportion assigned to them [by our demonstrations]. GALILEO AND TORRICELLI 139 u As for the measurement of the time, a large bucket filled with water was suspended in the air. A small hole in its base allowed a thin stream of water to escape, and this was caught in a small receptacle throughout the duration of the ball's descent of the channel, or of portions of it. The quantities of water caught in this way were weighed on a very accurate balance. The differences and relations of these weights gave the differences and relations of the times with such accur acy that, as I have said, these operations never gave a noticeable differ ence when repeated many times. " Galileo then introduced the notion of impeto, which he also called talento and momenta del discendere. For a given body, this tendency to motion is greatest along the ver tical BA. It is less on the planes AD, AE, AF. Finally, the impeto is completely reduced to nothing on the horizontal CA, where the body (as we have seen in reading the Mechanics) " is indifferent to motion or to rest, and does not of itself show any tendency to move in any direction or any resistance to being set in motion. " Salviati gives the following explanation of this fact. " In the same way that it is impossible that a heavy body, or an ensemble of heavy bodies should, of its own accord, move upwards and thereby go further away from the common centre to which heavy things tend, so it is impossible that it should spontaneously move if its centre of gravity does not approach the common centre in its motion. There fore the impeto of the moving body will be nothing on some chosen horizontal, or on a surface which is equidistant from the aforesaid centre and is without inclination. " This is the Galilean form of the principle of inertia. Galileo arrived at it by a kind of limiting process, starting from the principle of virtual work. Galileo then returned to the demonstration which he had given of the law of heaviness on an inclined plane, and in which he had appealed to the principle of virtual work. He completed it, however, in the following way. " Manifestly, the resistance, or the smallest force which suffices to stop or prevent a heavy body in its descent, is as great as the impeto of that body. In order to measure this force, I shall make use of the gravity of another body. Imagine that a body G rests on the plane FA and that it is attached to a thread which passes over F and supports a weight H. . . . In the triangle AFC9 the displacement of the body G, for example upwards from A to .F, is made up of the transverse and horizontal motion AC and the vertical motion CF. Now the resistance to motion due to the horizontal displacement is zero. . . . Consequently 140 THE FORMATION OF CLASSICAL MECHANICS the resistance is solely due to the fact that the body must climb the vertical CF. Therefore the body G, moving from A to F, only resists because of the vertical elevation CF. But the other body, H , necessarily descends the whole length FA in a vertical direction. . . . We can there fore say that when equilibrium is established the moments of the bodies, their velocities or tendencies to motion, that is, the distances which they would travel in the same time, will be in inverse ratio to their Fig. 46 gravities in accordance with the law which is true in every instance of motion in mechanics. It follows that to prevent the fall of G, it will suffice that H should be so much lighter with respect to G as the distance CF is less than FA. ... And since we have agreed that the impeto of a moving body, the energy., the moment or the tendency to motion has the same size as the force, or least resistance, which suffices to keep it still, we conclude that the body H is sufficient to prevent the motion of the body G. . . . " We note that Galileo's fundamental idea consists in measuring the impeto, or the tendency to motion, by means of the static force which can be opposed to it. This was an essentially original procedure which had escaped the notice of all the Schoolmen. As Jouguet1 has legiti mately remarked, the same word impeto, in Galileo's work, sometimes meant the velocity acquired by a body in a given time, and sometimes the distances travelled on differently inclined planes in a certain time, starting from rest. By means of the preceding considerations Galileo verified that the postulate according to which the velocities of a body which starts from 1 L. M., Vol. I, p. 106. GALILEO AND TORRICELLI 141 rest and falls along the line of greatest slope on differently inclined planes of equal height are the same when it arrives at a given horizontal. He also showed that " if the same moving body falls, starting from rest, on an inclined plane and along the vertical with equal height, the dura tions of fall have the same relation as the lengths of the inclined plane and of the vertical. " This demonstration was necessary in order to give full weight to his experimental verification of the law of falling bodies. 3. GALILEO AND THE MOTION OF PROJECTILES. We have seen that the Schoolmen and those interested in mechanics in the XVIth Century had only been able to treat the motion of projec tiles very imperfectly. Galileo solved this problem by means of a very remarkable analysis in which, together with the principle of inertia, there appears the principle of the composition of motions or of the inde pendence of the effects of forces. "We shall quote from the text of the Discorsi. " The Text. — I imagine a moving body thrown on a horizontal plane without any obstacle. It is said that its motion on the plane will remain uniform indefinitely if the plane extends to infinity. But if the plane is limited, and if it is set up in air, when the body, which we suppose to be under the influence of gravity, passes the end of the plane it will add to the first uniform and indestructible motion, the downward propensity which it has because of its gravity. From this will arise a compound motion, composed of the horizontal motion and the naturally accelerated motion of descent. I call this kind of motion, projection. " Animated by the motion composed of a uniform horizontal motion and a naturally accelerated falling motion, the projectile describes a parabola. " Let there be a horizontal or a horizontal plane, AB, which is placed in air and along which a body moves uniformly from A to B. At B, where its support is missing, the body, because of its weight, is forced by its gravity into a natural downward motion along the vertical BN. Produce AB into the line BE, which we shall use to measure the passage of time. Mark off equal lines BC, CD, DE on BE, and draw parallels to BN through the points C, D, E. On the first of these paral lels take an arbitrary length CI ; on the next one, a length DF which is four times as great ; on the third, a length EH nine times greater ; and so on, the successive lengths increasing as the squares of CB, DB, EB. . . . Imagine that the vertical descent along CI is added to the displacement of the body as it is carried from B to C in uniform motion. 142 THE FORMATION OF CLASSICAL MECHANICS At the time BC the body will be at I. At the time BD, which is twice BC, its vertical distance of fall will be equal to 4CI. For it has been proved that the distances are as the squares of the times in naturally accelerated motions. In the same way, the distance EH that is tra velled in the time BE will be nine times CI, so that the distances Eff, DF, CI are related to each other as the squares of the lines EB, DB, CB. . . . The points I, F, H therefore lie on a parabola. " D Fig. 47 The discussion between the three characters in the dialogue is of considerable interest. Sagredo remarks that the argument supposes that the two motions combined in this way " neither alter each other, nor confuse each other, nor mutually hinder each other in mixing up. " He objects that, since the axis of the parabola is vertical and goes through the centre of the earth, the particle will be separated from this centre. . . . Simplicio reproaches the text for, in the first place, neglec ting the convergence of the verticals and, in the second, neglecting the resistance of the medium. Salviati replies that, to a first approximation, these objections may be dismissed. He has experimented on a ball of wood and one of lead which were arranged to fall from a height of 200 cubits. The wooden ball, which was more sensitive to the resistance of the air, was not noticeably retarded. Salviati recalls that the projectiles from firearms have such velocities that their trajectories can be modified by the re sistance of the air. GALILEO AND TORRICELLI 143 4. GALILEO AND HYDROSTATICS. Galileo took up the study of hydrostatics in a manuscript called Discorso intorno alle cose che stanno in su Vacqua o che in quella si muo- vono. This was published at Florence in 1612. Essentially, his hydro statics was based on the principle of virtual velocities, which was directly inspired by Aristotle's mechanics. In this work, Galileo called the product of the force and the velocity, momenta. " I borrow two principles from the Science of mechanics. The first is this — two absolutely equal weights that are moved with equal velo cities are of the same power, or the same momenta, in all their doings. " To students of mechanics, momento means that property, that action, that efficient power by which the motive agency moves and the body resists. This property does not only depend on the simple gravity, but also on the velocity of motion, the different inclinations and the different distances travelled. Indeed, a heavy body produces a greater impeto when it descends on a very steep surface than when it descends on a surface which is less steep. Whatever may be the ultimate cause of this property, it always keeps the name momento. " The second principle is that the power of the gravitation increases with the velocity of the thing that is moved, so that absolutely equal weights that are animated with unequal velocities have unequal powers, strengths, unequal momenti. The more rapid is the more powerful, and this in the ratio of its own velocity to the velocity of the other weight. . . . " Such a compensation between the gravity and the velocity is found in all machines. Aristotle has taken it as a principle in his Pro blems of Mechanics. Hence the assertion, that two weights of unequal size are in equilibrium with each other, and have equal momenti, when ever their gravities are in inverse ratio to the velocities of their motion, may be taken as wellestablished. " In discussing the siphon, Galileo remarked that a small mass of water contained in a narrow vessel could maintain in equilibrium a large mass of water contained in a wide vessel, because a small lowering of the second entailed a great increase in the height of the first. In this respect Galileo preceded Pascal. If Duhem is to be believed, Galileo was guided by a tradition that went back to Leonardo da Vinci.1 The Discorsi were attacked by L. della Colombe and V. di Grazia, and defended by Benedetto Castelli (1577-1644), a faithful disciple of Galileo. The same Castelli was the author of a treatise on the measu rement of running water (Della misura delVacque correnti, 1628) which repeated Leonardo da Vinci's law of flow, Sv = constant. 1 fitudes sur Leonard de Vinci, Vol. II, p. 214. 144 THE FORMATION OF CLASSICAL MECHANICS Further, Galileo related the properties of the equilibrium of floating bodies to the principle of virtual velocities. Like his contemporaries, Galileo also believed in the horror vacui (resistenza del vacuo) . However, it is reported that he was very sur prised to learn that a newly constructed pump, whose aspiration tube was very long, could not lift water higher than eighteen Italian ells. Therefore he believed that this height implied a kind of ceiling to the horror of the vacuum. In addition, Galileo attempted to determine the weight of air by weighing a balloon that was filled with air, then heated in order to partially expel the air, and weighed again. As Mach has remarked, it is very true that the heaviness of air and the horror vacui were quite separate concepts before Pascal's time.1 5. GALILEO AND THE COPERNICAN SYSTEM. We shall briefly summarise Galileo's astronomical work. By means of a lunette which he had had constructed at Venice in 1609, he disco vered the satellites of Jupiter on Jan 7th, 1610 and observed that they accompanied the planet in its annual motion. This suggested the same possibility for the Moon in relation to the Earth. On the other hand, he noticed the phases of Venus and the sunspots, and thus obtained proof of the rotation of these two stars which was of first importance for supporting the hypothesis of the Earth's rotation. Finally he demonstrated a libration in the Moon's longitude. He was forced to retract his views on the Earth's rotation when he was first accused by the Inquisition in 1615. Nevertheless Galileo hastened to publish, at Florence in 1632, Four Dialogues on the two principal systems of the World, those of Copernicus and Ptolemy. (This in spite of the fact that he usually hesitated about printing his work because of his shortage of money ; even though he was content to distribute a few copies of the Discorsi among his friends in 1636.) The three speakers that will later appear in the Discorsi, Simplicio, Sagredo, and Salviati also appear in these dialogues. Galileo applied a searching dialectic to the scholastic arguments, here expressed by Simpli cio. For example, in Dialogue II, Simplicio enumerates the scholastic axioms, such as the unity of the cause and the unity of the effect, the necessity of an extrinsic source for all motion, natural or otherwise. These axioms conflict with the triple motion of Earth which Copernicus has suggested. This triple motion comprises the diurnal motion, the annual motion and the displacement of the Earth's axis parallel to itself. 1 MACH, AT., p. 106. GALILEO AND TORRICELLI 145 (Rather oddly, Copernicus had believed this to be one of the modes of the Earth's motion.) Salviati replies to this by assembling the experi mental evidence. And if he dares to contradict Aristotle, it is because the telescope has made the eyes of the astronomer thirty times more powerful than those of the philosopher. " Jam autem nos, beneficio Telescopii, tricies aut quadragies propius quam Aristoteles admovemur Caelo, sic ut in eo plurima possumus observare quae non potuit Aristoteles et, inter alia, maculas istas in Sole, quae prorsus ei fuerunt invisibiles. Ergo de Caelo, deque Sole, nos Aristotele certius tractare possumus. " x In his third dialogue Galileo concludes that though the copernican system may be difficult to visualise, it is simple in its effects. " Systema Copernicanum intellectu difficile et effectu facile est. " It is reported that this work brought Galileo a denunciation from the Holy Office, which obliged him to renounce his copernican beliefs and to remain in compulsory residence at Arcetri, near Florence. Here he died, surrounded by a number of disciples. Among these was Torricelli, who had only belonged to the circle for a few months. 6. TORRICELLI'S PRINCIPLE. We know that Galileo had already related the problem of the inclined plane to the principle of virtual work and that he had maint ained, in his Discorsi, that an ensemble of heavy bodies could only start to move spontaneously if its centre of gravity came nearer to the common centre of heavy things. Torricelli made this remark precise, and raised it to the status of a principle, in his treatise De Motu gravium naturaliter descendentium et projectorum (Florence, 1644). " We shall lay down the principle that two bodies connected together cannot move spontaneously unless their common centre of gravity descends. " Indeed, when two bodies are connected together in such a way that the motion of one determines that of the other, this connection being produced by means of a balance, a pulley or any other mechanism, the two bodies will behave as a single one formed of two parts. But such a body will never set itself in motion unless its centre of gravity falls. But if it is made in such a way that its centre of gravity cannot fall, the body will certainly remain at rest in the position that it occupies. From another point of view, it would move in vain because it would take a horizontal motion which did not tend downwards in any way. " 1 We have quoted a Latin edition which appeared at Lyons in 1641. 10 146 THE FORMATION OF CLASSICAL MECHANICS Torricelli applied this principle to two bodies on differently inclined planes and attached to each other by a weightless thread. Similarity, he applied it to the balance. All these examples are instances of indifferent equilibrium. If f is the height of the centre of gravity, reckoned algebraically on an ascending vertical, Torricelli's principle may be written (5C-0 for all virtual displacements compatible with the constraints. But the examples which he gave were all of the type <Jf = 0. Torricelli's true merit is not so much that of having won this principle from Galileo's mechanics, but that of having specified that the verticals should be treated as parallel. At the same time he renounced the scholastic conception of a common centre of heavy bodies at a finite distance, where the verticals converged. He writes, " This is an objection that is very common among the most thoughtful authors — Archimedes has made a false hypothesis in regarding the threads that support the two weights hung from a balance as being parallel to each other — in reality, the directions of these two threads meet at the centre of the Earth. . . . " The foundation of mechanics which Archimedes adopted, namely the parallelism of the threads of a balance, may be deemed false when the masses hung from the balance are real physical masses, tending towards the centre of the Earth. It is not false when these masses, whether they be abstract or concrete, do not tend towards the centre of the Earth, or to any other point near the balance, but towards some point which is infinitely distant. " We shall continue to call this point, towards which masses hung from the balance tend, the centre of the Earth. " Beneath these verbal precautions, and in spite of the fact that he did not refer to the orders of magnitude, Torricelli's intention of treating the verticals as parallels is clear. In Torricelli's principle the word " descend " is intended to indicate a tendency towards a centre which is taken to infinity, 7. TORRICELLI AND THE MOTION OF PROJECTILES. Galileo fully discussed the parabolic motion of a projectile which was thrown horizontally. Only in passing did he remark that, if a projectile was thrown obliquely from the point B with a velocity equal GALILEO AND TORRICELLI 147 and opposite to that with which it arrived at B after having been thrown horizontally from A, it would describe the same parabola in the opposite direction. Galileo made use of this appeal to an inverse return without proof. Moreover, he announced that the greatest range for a given velocity, if the projectile was thrown from JB, was obtained when the trajectory at B made half a right- angle with the horizontal. In this matter too, Torricelli systematised Galileo's work. Thus, in Book II of his De motu gravium, he considered a body that was projected obliquely. He compounded the uniform velocity in the direction of the velocity of projection with the accelerated motion. . ,.,. Gassendi had studied the same problem <-> ^ as early as 1640, in a treatise Tres Epistolae de motu impresso a motore translate. He considered a body projected upwards from the deck of a ship in uniform motion, and showed that the trajectory was a parabola. 8. TORRICELLl'S EXPERIMENT. No doubt inspired by Galileo's researches on the resistenza del vacua, Torricelli was lead to make experiments on a column of mercury rather than a column of water. The classical experiment with which his name is still associated was, however, accomplished by Viviani in 1643. 9. TORRICELLl'S LAW FLOW THROUGH AN ORIFICE. Torricelli seems to have been the inventor of hydrodynamics. Thus he observed the flow of a liquid through a narrow orifice near the bottom of a vessel. Dividing the total duration of flow into equal parts, he established that the quantities of liquid caught by some suitable receptacle increased regularly, from the last interval of time to the first, and that they were proportional to the odd numbers taken consecutively. This analogy with the law of falling bodies induced him to investigate the height to which the water that flowed out of the orifice could rise, if suitably directed upwards. He established that this height was always less than that of the liquid in the vessel. Moreover, he supposed that the stream would attain this height if 148 THE FORMATION OF CLASSICAL MECHANICS the resistances did not exist. TorricelE then formulated the law that the velocity of the liquid flowing out of the orifice was proportional to the square root of the height of the liquid. This statement was obtained by analogy with the motion of heavy bodies and was given without proof. It attracted the attention of Newton and Varignon, and thus lies at the bottom of the first investigations in hydrodynamics. CHAPTER THREE MERSENNE (1588-1648) AS AN INTERNATIONAL GO-BETWEEN IN MECHANICS ROBERVAL (1602-1675) 1. THE ARRIVAL OF FOREIGN THEORIES IN FRANCE. THE PART PLAYED BY MERSENNE. In 1634 there appeared simultaneously, in French, the translation of Stevin's mathematical work, P. Herigone's Cours Mathematique and Mersenne's translation of Galileo's Mechanics. Herigone's Cours Mathematique was inspired by Stevin's work, and took over the proof concerning equilibrium on an inclined plane. However, a column of liquid was unhappily substituted for Stevin's necklace of spheres. Herigone also borrowed many things from Guido Ubaldo and the statics of Jordanus and his school — in particular, the solution of the problem of the inclined plane. In this he was helped by the Italian Renaissance and the tradition — which we have discussed in connection with Tartaglia — that honoured Jordanus* contribution to statics. Though an omniverous reader, Father Mersenne (1588-1648) did not thereby arrive at a synthesis of his material. However, he established contact between the great students of mechanics, to whom he was continually posing questions, providing references and transmitting replies. His correspondence is like an international review of mechanics. Mersenne's Synopsis mathematica (1626) reviewed the work of Archimedes, Luca Valerio, Stevin, Guido Ubaldo and many others. As early as 1634 he translated Galileo's Mechanics. He told of Galileo's first work on the fall of bodies in Harmonicorum libri (1636), and added to this a traetise on mechanics by Roberval. He also made the work of Benedetto and Bernardino Baldi known to french students of the subject. In 1644 he published another compilation under the title Tractatus mechanicus. 150 THE FORMATION OF CLASSICAL MECHANICS Muck original work has only been preserved for us in the form of letters to Mersenne. In a time when authors were not liberal of refe rences, and disposed to pass of their writings as entirely original, Mer- senne's self-imposed task of liason and dissemination was quite essential. 2. ROBERVAL AND COMPOUND MOTION. We cannot describe the work in kinematics in the XVIIth Century, for an analysis of this would more properly belong to a history of geometry. Nevertheless, Roberval's kinematic geometry deserves a special mention because he was able to solve the problem of drawing tangents to different curves — a preoccupation among geometers of the time — by means of the composition of velocities. The Treatise on compound motion was only published by the Academic des Sciences in 1693. It was edited by a gentleman of Bordeaux on the basis of Roberval's lectures, and the latter confined his own con tribution to the addition of marginal notes. The essential principle used in this treatise is the following one. " Using the particular properties of the curved line that will be given to you, examine the different motions that a point which describes the line can have in the neighbourhood of the point at which you wish to find the tangent. Of all these component motions, take the line of the direction of the compound motion. You will have the tangent to the curved line. " For example, in the curve described by a point M fixed on a circle which rolls without sliding on a straight line, Roberval compounds an elementary translation of the base with an elementary rotation of the circle. This leads to a direction of compound motion which is perpendicular to the straight line joining M to the point of contact of the circle with the base. Roberval treates fourteen examples in the same way — for instance the cycloid, the conchoid, Archimedes' spiral and the conies, and he succeeds in drawing their tangents correctly. How ever, it turns out that he begs the question by giving the components of the velocity without precise justification. Descartes was to replace this by a method that became that of the instantaneous centre of rotation. 3. ROBERVAL'S TREATISE ON MECHANICS. In 1650 Roberval wrote to Hevelius, " We have constructed a new mechanics on the foundation already laid. Except for a small number, the ancient stones with which it has previously been constructed have been completely rejected. It consists of eight stages, corres ponding to a similar number of books. " MERSENNE AND ROBERVAL 151 The Bibliotheque Nationale (Paris) has a manuscript (No. 7226) which is undoubtedly an outline of this work. Even though he denies this, the mechanics which the author — " in the chair of Ramus " at the College of France — contemplates is most often inspired by Aristotle and the Italian Renaissance. Roberval claims to have only read Archimedes, Guido Ubaldo and Luca Valerio. But he is clearly subject to Baldi's influence. For example, this is what Roberval writes on the motion of projectiles. " The violence of a cannon-shot is made up of two impressed [motions]. One is purely violent, arising from the cannon itself and from the powder which is ignited to drive the shot along. The other is natural, being caused by the shot's own weight. Of the first impress ion, the violence increases somewhat at some distance from the cannon because of the degrees acquired by the motion, which are added to the impression of the powder before this has decreased appreciably. It then happens that, since the impression decreases much more in itself than it is added to by the degrees of velocity acquired, it conti nually slows down and, after a certain time, finishes. Now at the beginning the line of the direction of this violent impression is directed towards the place at which the cannon points. Later it changes conti nually and the cause of this change is the natural impression, that is, the body's heaviness carrying it towards the centre of the Earth. For the mixture of these two impressions, violent and natural, means that the shot does not exactly proceed along one direction or the other. But at the beginning it almost entirely follows the violent one, which is, without comparison, much greater than the natural one. Later the violent one disappears bit by bit, and so the shot begins to descend a curved line, and this all the more as the violent impression decreases and the natural motion is added to by the degrees acquired. " 4. ROBERVAL AND THE LAW OF COMPOSITION OF FORCES. Roberval's first claim to fame in statics is that of having justified the law of the parallelogram offerees. This he accomplished by starting from the condition for the equilibrium of the angular lever. We shall follow the treatise that Mersenne appended to his Harmonicorum libri. Roberval's work is more modest than the one to which we have just referred, and only occupies 36 pages. We shall analyse Roberval's demonstrations instead of quoting them — their style is heavy and artificially complicated, while their basis is simple. In the first place, Roberval considers a weight P suspended at B by 152 THE FORMATION OF CLASSICAL MECHANICS two strings AB and BC. The string A passes through the fixed point A. Roberval sets out to determine the traction Q which must be applied to the string BC in order to support the weight P. He replaces the arm AB, whose length is fixed, by an angular lever Ap, Aq, where Ap Q Fig. 49 is the perpendicular on the line of action of the weight P, and Aq is the perpendicular to the string BC. The equilibrium of an angular lever requires that P = Aq Q~ Ap' From this the value of Q is obtained. Roberval then applies this result to the next diagram. Here QG and CB are, respectively, perpendicular to CA and QA. Further, CF and QD are perpendicular to the line of action of the weight A. The weight A is suspended from the two strings CA, QA^ to which are applied the powers K and E. The equilibrium of the lever CF, CB A CB gives the ratio — = •— . Similarity, the lever ()D, Q G gives the ratio Jb CJT K~ QD " Therefore it is observed that in both cases two perpendiculars are drawn from each power — one on the direction of the weight and the other on the string of the other power. Also that, in the ratios of the weight to the powers, the weight is homologous to the perpendiculars falling on the strings of the powers. Similarly the powers are homolo gous to the perpendiculars falling on the direction of the weight. " MERSENNE AND ROBERVAL 153 By these purely geometrical considerations Roberval finally trans forms the statement of the preceding rule and arrives at the decompo sition of the weight into its two components in the directions of CA and QA. " If, from some point taken on the line of the direction of the weight, the line parallel to one of the strings is drawn to the other string, the sides of the triangle thus formed will be homologous to the weight and the two 'js interesting to remark that Roberval attempted to relate the rule of the composition of forces to the principle of virtual work. " In connection with a weight suspended by two strings, we have noticed a thing that has given us much pleasure. This is that when the weight is supported thus by two powers, it can neither rise nor fall without the reciprocal proportion of the paths with the weight and the two powers being changed, and this contrary to the common order. . . . " If a line AP is taken underneath A, in the line of its direction, it turns out that if the weight A falls as far as P, drawing the strings with it and making the powers rise, the reciprocal ratio of the paths that the powers travel in rising and the path which the weight travels in tailing will be greater than that of the same weight and the two powers taken together. Thus the powers will be raised further in the proportion that the weight descends in carrying them along, which is contrary to the common order. " . , An analogous argument is applied to the rising of the weight, ana this conclusion follows- " Consequently the weight A, in remaining in its place also remains in the common order. " CHAPTER FOUR DESCARTES' MECHANICS PASCAL'S HYDROSTATICS 1. DESCARTES' STATICS. Descartes9 statics stems directly from the principle of virtual work, which he assumed a priori. We shall quote a letter from Descartes to Constantin Huyghens dated October 5th, 1637. " The invention of all [simple machines] is only based on a single principle, which is that the same force that can lift a weight of, for example, a hundred pounds to a height of two feet, can also lift one of two hundred pounds to a height of one foot, or one of four hundred pounds to a height of half a foot, and so on, however this may be applied. " And this principle cannot fail to be accepted if it is considered that the effect should always be proportional to the action which is needed to produce it. So that if it necessary to use the action by which a weight of a hundred pounds can be lifted to a height of two feet, in order to lift some weight to a height of one foot, this weight should weigh two hundred pounds. For it is the same to lift a hundred pounds to a height of one foot, and then again, to lift a hundred pounds to the same height of one foot as to lift two hundred pounds to a height of one foot and also the same as to lift one hundred pounds to a height of two feet. " Now the machines which serve to make this application of a kind that acts on a weight over a great distance, and makes this rise by a smaller one, are the pulley, the inclined plane, the wedge, the lathe or turner, the screw, the lever and some others. For if it is not desired to relate some, they could be further enumerated. And if it is desired to relate them in such a way, there is no need to put down as many. " If it is desired to lift a body F, of weight 200 pounds, to the height of the line BA, in spite of the fact that the force is is only sufficient to lift one hundred pounds, it is only necessary to drag or roll the body DESCARTES AND PASCAL 155 F Fig. 51 along the inclined plane CL4, which I suppose to be twice as long as the line AB. For in order to bring it to the point A by this path, only the force which is necessary to make a hundred pounds rise twice as high would be used. . . . " But to be set against this calculation is the difficulty there will be in moving the body F along the plane AC if this plane had been laid along the line BC, whose parts I assume to be equally distant from the centre of the Earth. Since this obstruc tion will be less as the plane is harder, more even and more polished, it is a fact that it can only be expressed approximately and is not very considerable. Further, there is no need to consider that the plane AC should be slightly curved on account of the fact that the line BC is a part of a circle which has the same centre as the Earth . . . for this is in no way appreciable. " In Descartes' work on the lever, the resistance is always a weight hung from the lever, and the power is constantly perpendicular to the arm of the lever. G-uido Ubaldo had made use of this practical observa tion and Descartes followed him. We shall return to the text. " I have postponed speaking about levers until the end because, of all the machines, used to lift weights, this is the most difficult to explain. " Consider this — that while the force which moves the lever descends along the whole semicircle ABCDE, although the weight also describes the semicircle FGHIK it is not lifted the whole length of the line FGHIK, but only the length of the straight line FOK. So that the proportion that the force which moves the weight must bear to the heaviness of the weight should not be measured by the proportion of the two dia meters, but rather by the proportion of the greatest circumference to the smallest dia meter. " Moreover, consider that in order to turn the lever it is by no means necessary that the force should be as great when the lever is near A or near E as when it is near B or near D. The reason for this is that, there, the weight rises less, as it is easy to see. And to evaluate exactly what this force should be at each point of the curved line ABCDE, it is necessary to know that it acts in x^" 156 THE FORMATION OF CLASSICAL MECHANICS the same way as if drew the weight on a circular inclined plane. Also that the inclination at each of these points on the circular plane should be measured by that of the straight line which touches the circle at that point. " Not only did Descartes assert the principle of virtual work but — and in this regard his priority is certain — he indicated its infinitesimal character. " The relative weight of each body should be measured by the start of the movement which the power that maintains it can produce, rather than by the height to which it can rise after it has fallen down. Note that I say start to fall and not simply fall, because it is the start of the fall that must be taken care of. " In passing, we recall Descartes' contempt of his contemporaries and predecessors. Naturally Mersenne had drawn his attention to Galileo — here is Descartes reply. " And in the first place, concerning Galileo, I will say to you that I have never seen him, nor have I had any communication with him, and that consequently I could not have borrowed anything from him. Also, I see nothing in his books that causes me envy, nor anything approaching what I would wish to call my own. " It seems to me foolish to think of the screw as a lever — if my memory is correct, this is the fiction that Guido Ubaldo used. " To assert his independence of Galileo he wrote to Mersenne in the following terms. " As for what Galileo has written on the balance and the lever, it explains the quod itafit rather well, but not the cur ita fit as I have done with my Principle. " This shows that Descartes believed that a prin ciple that had been set up overrode all other considerations, even experi mental ones. . . . In the texts that we have quoted, Descartes continually uses the word force to denote what we now call work. Even in his own time some misunderstandings arose, and he was quick to take offence. On November 15th, 1638, he wrote to Mersenne on this matter. " At last you have understood the word force in the sense that I use it when I say that it takes as much force to lift a weight of 100 pounds to a height of one foot as to lift one of 50 pounds to a height of two feet. That is, that as much action or as much effort is needed. " Descartes clarifies this later (September 12th, 1638). " The force of which I have spoken always has two dimensions and is not the force which might be applied at some point to maintain a weight, which always has only one dimension. " Force in Descartes sense is therefore expressed by the product pi of DESCARTES AND PASCAL 157 a weight and a distance while the momenta in Galileo's sense 1 is express ed by the product pv of a weight and a velocity. Descartes formally claims to have excluded consideration of the velocity, " which would make it necessary to attribute three dimensions to the force. " He adds — " As for those who say that I should consider the velocity as Galileo has done I believe, among ourselves, that they are people who only talk nonsense and that they understand nothing in this matter. " Writing to Boswell in 1646, Descartes returned to this theme, which lay close to his heart. " I do not deny the material truth of what the students of mechanics are accustomed to say. Namely, that the greater the velocity at the end of the long arm of a lever is, in relation to the velocity at the other end, the less force it requires to be moved. But I deny that the velocity or the slowness are the causes of this effect. " Thus Descartes rejects all connection between statics and Aristotle's dynamics — of which traces subsist even in some of Galileo's concepts. Statics is made to depend on a single principle, which he asserts to be an obvious reality. Writing to Mersenne on September 12th, 1638, he said, " It is impossible to say anything good concerning the velocity without having to explain what heaviness is, and, in the end, the whole system of the World. " With regard to Roberval, who had claimed Mersenne's recognition of his own priority in connection with the postulate of statics that Descartes used, the latter shows himself to be even more contemptuous. " I have just read your RobervaPs Treatise on Mechanics, in which I learn that he is a professor — something I did not know. ... As for his Treatise, I would be able to find a large number of mistakes in it if I wished to examine it carefully. But I will say to you that, on the whole, he has taken a great deal of trouble to explain a thing that is very easy, and that, by means of his explanation, he has made it more difficult than it naturally is. Stevin showed the same things before him, and in a much more facile and general way. It is true that I do not know whether either of them is correct in his demonstrations, for I cannot have the patience to read the whole of these books. When he claims to have included something in a Corollary that is the same as I have done in my Writing on Statics, aberrat toto Caelo, he is making something that I made a principle, a conclusion, and he talks of time and of velocity in places where I talk of distance. This is a very serious mistake, as I have explained in my earlier letters. " 2 This haughtiness had its inconveniences. For this refusal to 1 Cf. above, p. 143. 2 Letter to MERSENNE, October llth, 1638. 158 THE FORMATION OF CLASSICAL MECHANICS read Roberval entailed Descartes' ignorance of the law of the compo sition of forces. In fact, the quantity pi had been considered as a measure of the work done by a weight by Jordanus and by Descartes contemporaries, Roberval and Herigone. We can agree with Duhem that " Descartes gave statics the order and the clarity which are the very essence of his method, but there is no truth in Descartes' statics that men had not know before. Blind ed by his prodigious pride, he only saw the error in the work of his predecessors and contemporaries. " l 2. DESCARTES AND THE FALL OF HEAVY BODIES. Descartes discussed the fall of bodies with Isaac Beeckman during his first stay in Holland (1617-1619). The fragment that we are going to analyse dates from this time, but Descartes returned to the subject in a letter to Mersenne dated November 16th, 1629. Descartes starts by recalling that a body which falls from A to B and then from B to C travels much more quickly in BC than in AB, " for it keeps all the impetus by means of which it moves along AB and besides, a new impetus which accumulates in it because of the effect of the gravity, which hurries it along anew at each instant. " This is the scholastic doctrine on the accumulation of impetus. " The triangle ABCDE shows the propor tion in which the velocity increases. " The line 1 denotes the strength of the impressed velocity at the first moment, line 2, the strength of the velocity impressed at the second moment, etc. . . . Thus the triangle lg4 ABE is formed and represents the increase of the velocity in the first half of the distance which the body travels. As the trapezium BCDE is three times greater than the triangle ABE, it follows that the weight falls three times more quickly from B to C than from A to B. That is, that if it falls from A to B in 3 moments, it will fall from B to C in a single moment. Thus in four moments its path will be twice as long as in three ; in twelve, twice as long as in nine ; and so on. " 2 10. S., Vol. I, p. 351. 2 CEuvres completes, Vol. I, p. 69. DESCARTES AND PASCAL 159 Therefore Descartes, like Galileo in 1604, assumed the law v=ks. Before he developed his own analytical geometry, he used the geome trical representation of uniformly varying quantities that was due to Oresme. Descartes called the measure of such a quantity the augmen- tatio velocitatis. But he confused the augmentatio along AB with the mean velocity along AB, which lead to a conclusion that was not only incorrect, but also in contradiction with the law from which he had started. Isaac Beeckman's Journal, which was used by Adam and P. Tannery in their edition of Descartes' works, contains further details of these discussions.1 Beeckman assumes the correct law v=kt and correctly deduces from it the law of distances. If AD represents a duration of one hour, the distance travelled is represented by the triangle ADE. In two hours the distance is represented by the triangle ADC. The ratio of the areas, and therefore that of the distances, is therefore the squared ratio of the times. Beeckman makes use of the method of indivisibles in order to justify this result. " If, during the first moment of time, the body has travelled a moment of distance AIRS, during the first two moments of time it will have travelled 3 moments of distance, represented by the figure AJTURS. The distance travelled in any time whatever is therefore represented by the corresponding triangle supplemented by the small triangles ASR, RUT, etc. . . . which are equal to each other. But these equal triangles added in this way are smaller as the moments of distance are smaller. Therefore these added areas will be of zero magnitude when it is supposed that the moment is of magnitude zero. It follows that the distance which the thing falls in one hour is to the distance through which it falls in two hours as the triangle ADE is to the triangle ACB. " The two propositions which Dominic Soto has stated are thus linked with each other by the bond of indivisibles. Beeckman ascribed even this argument to Descartes. " Haec ita de- monstravit Mr. Peron. " Unfortunately Beeckman did not persist in this point of view. In another writing 2 he went back to the law v = ks and repeated 1 (Euvres completes, Vol. X, p. 58. 2 Ibid., Vol. X, p. 75. Fig. 54 160 THE FORMATION OF CLASSICAL MECHANICS the very same error as Descartes in the evaluation of the mean velocity. In spite of the efforts which Duhem made to elucidate them,1 these essays remain somewhat confused. In this matter of the fall of heavy bodies, it remains that Descartes' contribution was not lasting and much less than that of Galileo, whose progress from 1604 to 1638 was conti nuous. Moreover, Galileo had a respect for observation, which Des cartes eschewed. 3. DESCARTES AND THE CONSERVATION OF QUANTITIES OF MOTION. As early as 1629 Descartes, writing to Mersenne, was categorical on the indestructibility of motion. " I suppose that the motion that is once impressed on a body remains there forever if it is not destroyed by some other means. In other words, that something which has start ed to move in the vacuum will move indefinitely and with the same velocity. " In his Dioptrics Descartes fell back on a mechanical model to ex plain the laws of reflection. A ball impelled from A to B bounces off the earth CBE. He explicitly neglects " the heaviness, the size and Fig. 55 the shape " of the ball, and supposed the earth to be " perfectly hard and flat. " He asserts that on meeting the earth the ball is reflected, and the " determination to tend to B which it had " is modified " with out there being any other alteration of the force of its motion than this. " In this connection, but in passing, he denied the theory of intermediate rest, which was dear to the hearts of some of the Schoolmen. He 1 DUHEM, Etudes sur Leonard de Kinci, Vol. Ill, p. 566. DESCARTES AND PASCAL 161 deemed " the determination to move towards some direction, like the movement, to be divided into all the parts of which it can be im agined that it is composed. " The ball is thus animated by two " deter minations. " One makes it descend and the other makes it travel horizontally. The impact with the ground can disturb the first but can have no effect on the second. Combining these principles with that of the conservation of the force of the motion of the ball, Descartes explained the laws of reflection. In his Principles (1644), Descartes reasserts the conservation of motion in a very detailed way, making it part of a metaphysical system. " God in his omnipotence has created matter together with the motion and the rest of its parts, and with his day-to-day interference, he keeps as much motion and rest in the Universe now as he put there when he created it. ..." By motion, Descartes understands what we now call quantity of motion, however precise his ideas on mass may be. 64 When a part of matter moves twice as quickly as another that is twice as large, we ought to think that there is as much motion in the smaller part as in the larger. And that each time the motion of one part decreases, that of some other part is increased proportionally. " Further, Descartes asserts the relativity of motion. " We would not be able to understand that the body AB is moved from the neighbourhood of the body CD if we did not also know that the body CD is moved from the neighbourhood of the body AB. No difficulty is created by saying that there is as much motion in one as in the other. " Moreover, he distinguishes between the proper motion of a body and " the infinity of motions in which it can participate because it is part of other bodies which move differently. " In order to illustrate this, he gives the example of the watch of a sailor that takes part in the motion of his vessel. Again, Descartes affirmed that the motion is conserved in a straight line : " Each part of nature, in its detail, never tends to move along curved lines, but along straight lines. This rule . . . results from the fact that God is immutable and conserves motion in nature by a very simple operation ; for he does not conserve it as it might have been some time previously, but as it is at the precise moment he conserves it. " And Descartes here recalls the motion of a stone in a sling ; he points out that we cannot " conceive any curvature in the stone, " of that we are " assured by experience " for the stone leaves " straight from the sling. . . ; which makes manifest to us that any body that is moved in a circle tends unceasingly to recede from the centre of the circle it describes ; 11 162 THE FORMATION OF CLASSICAL MECHANICS and we can even feel this with our hand while we turn the stone in the sling, because it pulls and makes the string taut in its effort to recede directly from our hand. " 4. DESCARTES AND THE IMPACT OF BODIES. Descartes formulated the following rules for the impact of bodies. 1) If two equal bodies impinge on one another with equal velocity, they recoil, each with its own velocity. 2) If one of the two is greater than the other, and the velocities equal, the lesser alone will recoil, and both will move in the same direction with the velocity they possessed before impact. 3) If two equal bodies impinge on one another with unequal vel ocities, the slower will be carried along in such a way that their common velocity will be equal to half the sum of the velocities they possessed before impact. 4) If one of the two bodies is at rest and another impinges on it, this latter will recoil without communicating any motion to it. 5) If a body at rest is impinged on by a greater body, it will be carried along and both will move in the same direction with a velocity which will be to that of the impinging body as the mass of the latter is to the sum of the masses of each body. 6) If a body C is at rest and is hit by an equal body J5, the latter will push C along and, at the same time, C will reflect B. If B has a velocity 4 it gives a velocity 1 to C and itself moves backwards with velocity 3. This, as an example, is how Descartes justifies this rule. " It is necessary that either B will push C along and not be reflected, and thus transfer 2 units of its own motion to C ; or that it will be reflec ted without pushing C along and, as a consequence, that it will retain these 2 units of velocity together with the 2 that cannot be lost to it ; or, further, that it will be reflected and retain a part of these two units and that it will push C along by transferring the other part. It is clear that, since the bodies are equal and, consequently, that there is no reason why B should be reflected rather than that it should push C along, these two effects will be equally divided. That is to say that B will transfer to C one of these 2 units of velocity and will be reflected with the other. " 7) Descartes also formulated a seventh rule relating to two unequal bodies travelling in the same direction. We remark that Descartes' guiding idea was the conservation of the quantity of motion, m\v , in absolute value. This idea was to persist DESCARTES AND PASCAL 163 among the Cartesians until the resolution of the controversy about living forces that we shall come to much later. Nearly all Descartes' rules on impact are experimentally incorrect. We remark that he may have suspected this, without being very much disturbed, when he said — " It often happens that, at first, the observations seem to be at variance with the rules I have just described. But the reason for this is clear. For the rules presuppose that the two bodies B and C are perfectly hard and so separated from all others that there is no other near them which can help or hinder their motion. We see nothing of this kind in the World. " Jouguet has a very legitimate comment to make on the preceding declaration. *6 This passage is very characteristic of Descartes' thought. He could observe nature and argue accurately from his laws as well as any other. But he had the pretension of rebuilding everything in a rational way according to the principles of his philosophy. He considered that the source of certainty lay in thought alone. It is known that he did not wish to assume the principles that were accepted in geometry and physics — further, by an exaggeration of his system, he came to neglect observation. " * 5. THE DISCUSSION BETWEEN DESCARTES AND ROBERVAL ON THE CENTRE OF AGITATION. In his Exercitationes we know that Bernardino Baldi had introduced a distinction between the centre of gravity and the centre of violence, or the centre of accidental gravity.2 Mersenne, who had read Baldi's work, suggested to geometers that they should search for a solid that would have the same period of oscillation as a simple pendulum of given length. Descartes replied to him in a letter dated March 2nd, 1646.3 " [The] point of your letter — which I do not wish to postpone answering — is the question concerning the size that each body that is hung in the air by one of its extremities should have, whatever its shape may be, in order that it should carry out its comings and goings equally with those of a lead hung by a thread of given length. , . . The general rule that I give in this connection is the following one. Just as there is a centre of gravity in all heavy bodies, so there is also a centre of their agitation in the same bodies, when these are hung from some 1 L. AT., Vol. I, p. 90. 2 See above, p. 106. 8 CEuvres completes, Vol. IV, pp. 362-364. 164 THE FORMATION OF CLASSICAL MECHANICS part and move. Also, that all those bodies in which this centre of oscillation is equally distant from the point from which they are sus pended, execute their comings and goings in equal times, provided that the change of this proportion that can be produced by the air is excepted. " He returned to this question on March 30th, 1646.1 " In the first place, since the centre of gravity is so situated in the middle of a heavy body, the action of any part of the body, which could, by its weight, divert this centre from the line along which it falls, is prevented by another part which is opposed to the first and which has just as much force. From this it follows that, in descent, the centre of gravity always moves along the same line as it would take if it were alone, and if all the other parts of which it is the centre were taken away. Similarly, what I call the centre of agitation of a suspended body is the point to which the different agitations of all the other parts of the body are related, so that the force which each part has to make itself move more or less quickly than it does is pre vented by that of another which is opposed to it. From which it also follows, ex definitione, that this centre of agitation will move about the axle from which the body is suspended with the same velocity that it would have if the remainder of the body of which it is a part were taken away and, as a consequence, the same velocity as a lead hung from a thread at the same distance from the axle. " Here Descartes raised an analogy to the status of a principle, and drew all the logical consequences from it with his accustomed vigour and clarity. But he did not confine himself to this. He tried to determine the position of the centre of agitation by forming the quantity mv for each element, or the proportional quantity mr (where r is the distance from the axis). But he took no account of the direction of these velocities — he always considered v, not v — and related every thing to a plane passing through the centre of gravity and the axis of rotation. Roberval pointed out this error in a letter to Cavendish dated May, 1646.2 " The defect of [Descartes'] argument is that he considers only the agitation of the parts of the agitated body, forgetting the direc tion of the agitation of each of those parts. For the centre of gravity is the cause of its reciprocation from left to right. " Descartes replies 3 that Roberval was mistaken " in thinking that the centre of gravity contributes anything to the measure of its vibra- 1 CEuvres completes, Vol. IV, pp. 379-388. 2 Ibid., p. 400. 3 Ibid., p. 432. DESCARTES AND PASCAL 165 tions beyond what the centre of agitation does. For the word centre of gravity is relative to bodies that move freely or which do not move in any way. For those which move about an axle to which they are attached, there is no centre of gravity with respect to that position and the motion has only a centre of agitation. " Fig. 56 Without delaying ourselves further with this controversy in which Descartes appears as his usual peremptory self, we shall confine our selves to the question of a plane figure oscillating about an axis through O.1 As we have already indicated, Descartes' calculation starts by bringing each element of the figure, M, to M! along the arc MM' of centre 0. The motion is then supposed to correspond to a quantity of motion applied perpendicularly to OG at M'. On the other hand, Roberval supposes that nw is applied at the point I on OG. Only the component normal to OG matters, since the other is nullified by the fixity of the support. Thus Roberval arrives at a correct determination of the centre of oscillation 0' in the case of a sector of a circle oscillating about an axis passing through the centre. Morsenne's correspondence has established that Huyghens was in volved in the same question as early as 1646, at the age of seventeen. At the beginning, his attitude to the problem was determined by the Cartesian discipline and he did not emancipate himself from this until much later, when he solved the problem of the centre of oscillation in his Horologium oscillatorium (1673) by appealing to the principle of living forces. 1 C/. JotrcuET, L. M., Vol. I, p. 158. 166 THE FORMATION OF CLASSICAL MECHANICS 6. THE QUARREL ABOUT GEOSTATICS. We are now going to say a little about a controversy which has, at least, the interest of showing that traces of Scholasticism remained even in the most distinguished minds of the XVIIth Century. In 1635 Jean de Beaugrand announced to Galileo, Cavalieri, Castelli and those interested in mechanics in France, that he had found the law which determined how the weight of a body varied with its distance from the centre of the Earth— the weight was proportional to the distance. This result appeared in 1636 in his Geostatics. In a letter to Mersenne, Descartes denied this in the following terms. " Although I have seen many scpiarings of the circle, perpetual motions and many other would-be demonstrations which were false, I can nevertheless say that I have never seen so many errors united in one single proposition Thus I can say in conclusion that what this book on Geostatics contains is so irrelevant, so ridiculous and mistaken that I wonder that any honest man has ever deigned to take the trouble of reading it. I would be ashamed of that which I have taken in recording my feeling in this letter if I had not done so at your request. " In May, 1636, Fermat formulated a proposition which he called Propositio Geostatica and which he expressed in the following way. Let B be the centre of the Earth, BA a terrestrial radius and J5C a part of the opposite radius. Consider two bodies A^ and C which are placed at A and C. If the weight A is to the weight C as JBC is to JEL4, the two bodies are in e<pzilibrium. Fig. 57 He adds, " It is very easy to demonstrate this result by following in the steps of Archimedes. " Given this, Fermat deduces the following result from his geostatic hypothesis. // -e * Fig. 58 DESCARTES AND PASCAL 167 44 Wherever a body JV is placed between B and A, if the proportion of AB to BN is equal to the proportion of the weight JV to the power R which is applied at A, the weight JV will be kept in equilibrium by the power. Therefore the nearer a body approaches the centre of the Earth, the smaller is the power at A that is necessary to maintain it in equili brium. This, errors apart, coincides with Beaugrand's geostatic pro position. " x Fermat gave the following explanation to Mersenne, who had not indicated his accord with this strange proposition in which Fermat had applied the laws of a lever to longitudinal forces.2 44 Every body, in whatever place except the centre of the Earth it may be, and taken by itself and absolutely, always weighs the same. . . . In my Proposition I never consider the body by itself, but only in relation to a lever, and thus there is nothing in the conclusions which is not included in the premisses. 44 Let A be the centre of the Earth, and let the body E be at the point E and the point JV be in the surface or somewhere else that is further away from the centre than the body E. I do not say that E weighs less when it is at E than when it is at JV. But I do say that if the body is suspended from the point JV by the thread JVE, this force at the point JV will support it more easily than if it were nearer to the said force, and this in the proportion that I have indicated to you. " Fermat was legitimately attacked by Et. Pascal and Roberval. Descartes also condemned him, and in this con nection, made clear his own ideas on heaviness in a Note of July 13th, 1638. 64 It is necessary to decide what is meant by absolute Fig. 59 heaviness. Most people understand it as an internal pro perty or quality in each of those bodies that are called heavy, which makes these bodies tend towards the centre of the Earth. According to some, this property depends on the shape, according to others, only on the material. Now according to these two beliefs, of which the first is most common in the Schools and the second most often accepted by those who can understand something out of the ordinary, it is clear that the absolute heaviness of bodies is always the same and that it does not change in any way because of their different distances from the centre of the Earth. 44 There is also a third belief — that of those who believe that there is no heaviness which is not relative, and that the force or property 1 (Euvres completes de Fermat, Vol. II, p. 6. 2 Ibid., p. 17, Letter of Fermat to Mersenne, 24th June 1636. 168 THE FORMATION OF CLASSICAL MECHANICS which makes the bodies that we call heavy descend does not lie in the bodies themselves, but in the centre of the Earth or in all its mass, and that this attracts them towards itself as a magnet attracts iron And according to these, just as the magnet and all other natural agents which have a sphere of activity are always more effective at small than at great distances, so it should be said that the same body weighs more when it is closer to the centre of the Earth. " For myself, I understand the true nature of heaviness in a sense that is very different from these three. . . . But all that I can say [here] is that by this I do not add anything to the clarification of the proposed question, x except that it is a purely factual one. That is to say that it can only be settled by a man in so far as he can make certain observa tions, and also that from the observations that are made here in our air it is not possible to know what there might be much lower down, towards the centre of the Earth, or much higher, among the clouds. Because if there is a decrease or increase of heaviness, it is not obvious that it follows the same proportion throughout. " Descartes was of the opinion that observation seemed to show that heaviness decreased as a weight was separated from the centre of the Earth. He gave strange evidence for this, such as " the flight of birds, the paper dragons that children fly and the balls of pieces of artillery that are fired directly towards the zenith and appear not to fall down again. " Another piece of evidence was that " since the planets which do not have light inside themselves, like the Moon, Venus and Mercury, are probably bodies of the same kind as the Earth, and since the skies are liquid as nearly all Astronomers of this Century believe, it seems that these planets would be heavy things and would fall towards the Earth if their great separation from it had not removed this inclination. " Returning to observation in the neighbourhood of the Earth, Des cartes considered the absolute heaviness as being practically constant. " If this equality in the absolute heaviness is supposed, it can be shown that the relative heaviness of all hard bodies, considered in the free air without any support, is somewhat less when they are near the centre of the Earth than when they are separated from it, though it may not be the same for liquid bodies. On the contrary, if it is supposed that two equal weights are opposed to each other on a perfectly accurate balance, when the arms of the balance are not parallel to the horizon, that one of the two bodies which is nearer the centre of the Earth will weigh more precisely to the extent that it is closer to it. To leave the example of the balance, it also follows that of the equal parts of the 1 The question is that of knowing whether a body weighs more or less when it is near the centre of the Earth than when it is further away. DESCARTES AND PASCAL 169 same body, the highest parts weigh more than than the lowest ones to the extent that they are further separated from the centre of the Earth, so that the centre of gravity cannot be fixed in any body, even a spher ical one. " As a general rule the convergence of the verticals renders consider ation of the centre of gravity illusory. Such was Descartes' conclusion. He also assumed, as Guido Ubaldo had already done, the law of attrac- k tion - (inversely proportional to the distance). This conclusion was important and, if one is to [believe Duhem,1 it stemmed from the Italian School — from Torricelli through the agency of Castelli. However, the Beaugrand-Fermat law of attraction kr (pro portional to the distance) allowed the existence of a fixed centre of gravity in a body. This had been shown by P. Saccheri in his Neo Stattica. 7. PASCAL'S HYDROSTATICS. Mersenne had advertised Torricelli's experiment in France as early as 1644. Pascal set about repeating the experiment with the collabo ration of Petit and convinced himself, by this means, of the possibility of a vacuum, " which Nature does not avoid with as much horror as many imagine. " In 1647 Pascal published New Experiments concerning the Vacuum. He gave full details of his plan for his great experiment in a letter to Perier dated November 15th, 1647, and the experiment itself was completed at the Puy de Dome — a mountain in Central France — on September 19th, 1648. The account of the experiment was published in October of the same year. The principal result was that the difference of level of mercury columns separated by a height of 500 toises was 3 pouces, one and a half lines. The Treatise on the Equilibrium of Liquids and the Heaviness of the Mass of Air appeared in 1663. In this work Pascal established that liquids " weigh " according to their height, and that in this respect a vessel of ten pounds capacity was equivalent to a vessel of one ounce capacity if both heights were the same. From this Pascal directly obtained the principle of the hydraulic press. " A vessel full of water is a new principle in mechanics, and a new machine for multiplying forces to whatever degree might be desired. " Pascal immediately relates this principle to that of virtual work. 1 0. S., Vol. II, p. 183. 170 THE FORMATION OF CLASSICAL MECHANICS ** And it is wonderful that in this new machine there is encountered the same constant order that is found in all the old ones, namely the lever, the windlass, the endless screw, . . . which is that the path increases in the same proportion as the force. ... It is clear that it is the same thing to make a hundred pounds travel a path of one pouce as to make one pound travel a path of a hundred pouces. " It is in this connection that Pascal defines the pressure — the water beneath both pistons of a hydraulic press is equally compressed. In Chapter II of the Treatise on the Equilibrium of Liquids there is mention of a " small treatise on mechanics, " now lost, in which Pascal had given " the reason for all the multiplications of forces which are found in all the instruments of mechanics so far invented- " This principle does not seem to have differed from Torricelli's prin ciple which Pascal used, without quotation, in hydrostatics. Fig. 60 " I take it as a principle that a body never moves because of its weight unless the centre of gravity descends. From this I prove that the two pistons in the diagram are in equilibrium of this kind. For their common centre of gravity is at a point which divides the line of their respective centres of gravity in the proportion of their weights. Now if they move, if this is possible, their paths will be related to each other as their reciprocal weights, as we have shown. Now if their com mon centre of gravity is taken in this second situation, it will be found in exactly the same position as previously. Therefore the two pistons, considered as one and the same body, move in such a way that their centre of gravity does not descend, which is contrary to the principle. Therefore they are in equilibrium. Q.E.D." The reason for the equilibrium in all Pascal's examples lies in the fact that " the material which is extended over the base of the vessels, from one opening to the other, is liquid. " In mechanics this property which belongs to incompressible fluids of wholly transmitting a pressure has been called Pascal's Principle, We shall not detain ourselves further with Pascal's Treatise — which DESCARTES AND PASCAL 171 has become classical — nor with the suggestive forms that he gave the hydrostatic paradox. Stevin had anticipated some of these ideas. Pascal's contribution to physics was in marked contrast with Des cartes9 fragile conceptions in the same field. However, it had its con temporary opponents — thus Pascal was obliged to contend with Aristo telians like Father NoeL At first Pascal took various verbal precautions when he argued the futility of the horror of the vacuum. But eventually he became forthright in his conviction that this scholastic prejudice was absurd. " Let all the disciples of Aristotle gather together all the strength in the writings of their master and his commentators in order, if they can, to make these things reasonable by means of the horror of the vacuum. Except that they know that experiments are the true masters that must be followed in physics. And that what has been accomplished in the mountains reverses the common belief of the world that Nature abhors a vacuum ; it has also established the knowledge — which will never die — that Nature has no horror of a vacuum, and that the heaviness of the mass of air is the true cause of all the effects which have previously been attributed to this imaginary cause. " CHAPTER FIVE THE LAWS OF IMPACT (WALLIS, WREN, HUYGHENS, MARIOTTE) THE MECHANICS OF HUYGHENS (1629-1697) 1. THE MECHANICS OF WALLIS (1616-1703). In 1668 the Royal Society of London took the initiative of choosing for discussion the subject of the laws of impacting bodies. Wallis (November 26th, 1668) discussed the impact of inelastic bodies, while Wren (December 17th, 1668) and Huyghens (January 4th, 1669) dis cussed the impact of elastic bodies. Wallis' memoir should be set against the background of his other work in mechanics, which was the subject of a treatise Mechanic^ sive de Motu (London, 1669-1671). In this treatise Wallis considered the vis matrix and the resistantia, which were opposed to each other in every machine. He hesitated between the concept of memento in Galileo's sense, which could be expressed by the product pvs of the weight and the vertical component of the velocity, and the notion of moment, or the product ph of the weight and the height of fall. In the first Chapter, Wallis even went as far as to consider a mixed solution by the product pvs as a measure of the momentum of the motive force and the product ph as a measure of the impedimentum of the resistance. Very fortunately he did not adhere to this peculiar choice and declared himself in favour of the product ph in the second Chapter of his Treatise, which was concerned with the fall of bodies. He went further and generalised the principle that Descartes had laid at the foundation of statics by extending it to forces other than the heaviness. " In an absolutely general way, the progress effected by a motive force is measured by the movement effected in the direction of this force, the recoil by the movement in the opposite direction. . . . The progress or recoil effected under the action of any force is obtained by WALLIS, WREN, HUYGHENS, MARIOTTE 173 taking the products of the forces by the lengths of the progress or recoil reckoned in the line of direction of the force. " Here Wallis supposes the displacements to be finite and rectilinear and the forces to be constant in magnitude and direction. But he assumes that a curvilinear trajectory can be represented as a limit of its tangents. It is said that Wallis also exploited the notion of work in all its generality, in so far as displacements due to a constant force were concerned. Without delaying ourselves over-much with this treatise, we shall discuss Wallis's treatment of the laws of impact. Wallis called a body perfectly hard if it did not yield in any way in impact. This is a category that must be distinguished from soft and elastic. " A soft body is one that yields at impact in such a way as to lose its original shape, like clay, wax, lead. . . . For these bodies part of the force is used to deform them — the whole force is not expended in the obstacle. It is necessary to take account of this part. " Like Jouguet,1 we remark that from the point of view of energy, Wallis had good reason to make this distinction. According to whether the internal energy of a body depends on its deformation or not, the living force lost in the impact is not equivalent, or is equivalent, to the heating of the body. But from the point of view of the quantity of motion, which was the one that Wallis took, this energetic distinction is irrelevant. Mariotte abandoned it, and was followed by others. Wallis called a body elastic if it yielded in impact, but then sponta neously regained its original shape, like a steel spring. Finally, he defined the direct impact of two bodies in the way that is now accepted. We now come to Wallis's proposition relating to soft impact (Chapter XI, Proposition II). " If a body in motion collides with a body at rest, and the latter is such that it is not moving nor prevented from moving by any external cause, after the impact the two bodies will go together with a velocity which is given by the following calculation. " Divide the momentum furnished by the product of the weight and the velocity of the body which is moving by the weight of the two bodies taken together. You will have the velocity after the impact. " Indeed, let a body A be in motion along the line AA that passes through its centre of gravity and through that of the body B which is at rest. Let p and v be the weight and the velocity of A. The impelling force (vis impellens) will be pv. Let p' be the weight of the body B 1 L. M.9 Vol. I, p. 123. 174 THE FORMATION OF CLASSICAL MECHANICS whose velocity is nothing. The weight of the two bodies is p + p'. After the impact the two bodies will move with the same velocity. Indeed, JB cannot go more slowly than A9 since A follows it. Neither can it go more quickly, for it is supposed that there is no other cause of motion than that which arises from the impulsion of A. (If there is another force, like the elastic force, which can impell B more quickly, the problem is of another order, to which we shall return.) Therefore the weight p + p' is moved by a force pv and its velocity is — — — -. " P + P This demonstration is based on the conservation of the total quantity of motion of the system. It has already been remarked that Wallis did not distinguish between the weight and the mass. Fig. 61 Wallis generalised this argument to the situation in which J5 is in motion with a smaller velocity than that of A, but in the same direction. If vf is the velocity of JS, the common velocity of the two bodies A and B after the impact is pv + p'v' This is always obtained by dividing the sum of the moments by the sum of the weights. If the velocity of B is in the opposite direction to the velocity v of A, and is denoted by — i/, the common velocity of the two bodies A and B after the impact is pv — p'vf P+P' ' It is seen that Wallis, unlike Descartes, was careful to take account of the signs of the quantities of motion. That is why he arrived at rules which were, apart from the confusion of the weight and the mass, correct. Finally, Wallis remarked that " the magnitude of the impact is equal to twice the decrease that is experienced by the greatest moment in direct impact. " Indeed, " Consider the body which has the greatest moment as hitting, and the other body as being hit. The body which is hit receives WALLIS, WREN, HUYGHENS, MARIOTTE 175 as much moment as is lost by the body which hits it. These moments that are gained or lost are both the restdt of the impact. The impact is therefore equal to their sum, that is, to twice the decrease experienced by the greater moment. " Wallis was also concerned with elastic impact (in Chapter XIII of his Treatise). He related this to the theory of soft impact. He introduced an elastic force (vis elastica) whose nature he did not specify, confining himself to an appeal to experimental facts. He stated the following proposition. " If a body hits an obstacle directly, and if the two bodies — or only one of them — are elastic, the first body will rebound with a velocity equal to that which it had before the impact, and will follow the same direction. " Indeed, " if the elasticity were nothing, the body A would come to rest. " — (This conclusion is obtained by applying the theory of soft impact to an immovable obstacle B. Moreover, we add that Wallis extended this result to an obstacle whose force of resistance was limited by comparing this force with a moment greater than that of the body A.) Wallis continues — 44 Therefore all motion remaining after the impact is the result of the vis elastica. Now this is always equal to the force of the impact. . . . Indeed, the elasticity does not resist as a simple impedimentum, but rather as a contrary force acting by reaction and with the same energy that the compression requires. Now what the elasticity suffers during the compression is equal to the impact. The restitutive force is there fore equal to the impact. . . . Now, in particular, since the body A has weight p and velocity t;, the magnitude of the impact is 2pv. This is also that of the elastic force. Since it is developed equally in the two parts half of this force, that is pv, acts on the obstacle and is dissipated there, while the other half repels the body A with velocity v. " Wallis also treats, for example, the elastic collision of two equal bodies which have equal and opposite velocities and — borrowing this from Wren — the collision of unequal bodies with velocities inversely proportional to their weights. Each body rebounds with the velocity that it has before the impact. These demonstrations are analogous to the preceding one and we shall not describe them here. 2. WREN (1632-1723) AND THE LAWS OF ELASTIC IMPACT. We shall describe the paper presented by Wren at the meeting in 1668 and which is included in the Philosophical Transactions of 1669. Wren starts from the concept of proper velocity which, for any body, 176 THE FORMATION OF CLASSICAL MECHANICS is inversely proportional to the weight. The impact of two bodies R and S which travel with their proper velocities results in the conserva tion of these velocities. If the velocities differ from their proper values, the bodies R and S " are brought back to equilibrium by the impact. " This is to say that if, before the impact, the velocity of R is greater than its proper velocity by a certain amount and that of S is less than its proper velocity by the same amount, as a result of the impact this amount is added to the proper velocity of S and subtracted from that of JR. Wren seems to have regarded the impact of two bodies with their proper velocities as equivalent to a balance oscillating about its centre of gravity. Wren expressed this analogy in the diagrams that he used to represent the effect of the impact. Strictly speaking, he did not justify his results in a logical and satisfactory way, but he had the merit of making experiments and of embodying his conclusions in a clear and precise law. 3. HUYGHENS (1629-1697) AND THE LAWS OF IMPACT. Following Wren's example, Huyghens confined himself to elastic impact. His researches were collected in a posthumous volume De Motu corporum ex percussione (1700). Huyghens' investigation was based on the following three hypo theses. 1) The first is the principle of inertia. " Any body in motion tends to move in a straight line with the same velocity as long as it does not meet an obstacle. " 2) The second is the following principle. Two equal bodies which are in direct impact with each other and have equal and opposite velocities before the impact, rebound with velocities that are, apart from sign, the same. 3) The third hypothesis asserts the relativity of motion. Huyghens shows himself to be a Cartesian in this matter. " The expressions 4 motion of bodies ' and ' equal or unequal velo cities ' should be understood relatively to other bodies that are consider ed as at rest, although it may be that the second and the first both parti cipate in a common motion. And when two bodies collide, even if both are subject to a uniform motion as well, to an observer who has this common motion they will repel each other just as if this parasitica1 motion did not exist. WALLIS, WREN, HUYGHENS, MARIOTTE 177 " Thus let an experimenter be carried by a ship in uniform, motion and let him make two equal spheres, that have equal and opposite velocities with respect to him and the ship, collide. We say [hypo thesis 2] that the two bodies will rebound with velocities that are equal with respect to the ship, just as if the impact were produced in a ship at rest or on terra firma. " Huyghens appealed to this relativity in order to justify the following proposition. " Proposition /. — If a body is at rest and an equal body collides with it, after the impact the second body will be at rest and the first will have acquired the velocity that the other had before the impact. " Imagine that a ship is carried alongside the bank by the current of a river and that it is so close to the edge that a passenger on the ship can hold the hands of an assistant on the bank. In his two hands, A and B, the passenger holds two equal bodies E and F which are hung from threads. Let the distance EF be divided into two equal parts by the point G. By displacing his two hands equally towards each other, the passenger will make the two spheres E and F collide with equal velocities with respect to himself and the ship (hypotheses 2 and 3). But during this time the ship is supposed to be carried to the right with a velocity GE equal to that with which the right hand of the passenger is moved towards the left. " Consequently the right hand, A, is motionless with respect to the bank and the assistant who is placed there, while the passenger's left hand, B, is displaced with a velocity EF — twice of GE or FG — with respect to the assistant. Suppose that the assistant placed on the bank grasps the passenger's hand A^ as well as the end of the thread which supports the globe jB, with his own hand C. Also, that with his hand D he grasps the passenger's hand B, which is the one that holds the thread from which the sphere F hangs. It is seen that when the passenger makes the spheres E and F meet each other with velocities that are equal with respect to himself and the ship, at the same time his assistant makes the sphere E — motionless with respect to himself and the bank — collide with the sphere jF whose velocity is FE. And it is certain that, if the passenger displaces the spheres in the way that has been described, there is nothing to prevent his assistant on the bank from seizing his hands and the ends of the threads, provided only that he accompanies the motion and does not oppose any hindrance to them. In the same way, when the assistant on the bank is directing the sphere F against the motionless sphere E, there is no obstacle to the passenger grasping his hands, even though the hands A and C are at rest with respect to the bank while the hands D and JB move with the same velocity EF. 12 178 THE FORMATION OF CLASSICAL MECHANICS " As we have seen, the spheres E and F rebound after the impact with velocities that are equal with respect to the passenger and the ship — that is, the sphere E with the velocity GE and the sphere F with the velocity GF. During this time the ship moves towards the right passenger's hand t's hand O F O Fig. 62 with the velocity GE or FG. Therefore, with respect to the bank and the assistant on it, the sphere F remains motionless after the impact and the sphere E moves to the left with a velocity twice GJ5 — that is, with the velocity FE with which F has hit E. We therefore show that, to an observer on the earth, when a motionless body is hit by an equal one, after the impact the second one loses all its motion which is, on the other hand, completely taken over by the other. " With the help of this remarkable artifice Huyghens treats all instances of the impact of two equal bodies by starting from the symme trical case, whose solution he assumes a priori. Thus he shows that two equal bodies that have unequal velocities exchange these velocities in a direct impact. He then passes to a consideration of unequal bodies and establishes the conservation of the relative velocity in the impact of two elastic bodies. It is of some interest to remark here that Huyghens showed, by examples, that the quantity of motion was not always conserved. In this context he was concerned with the quantity of motion m| v|, in Descartes' sense. Finally, he demonstrates Wren's rule on the conservation of proper velocities. " Proposition VIII. — If two bodies moving in opposite directions and with velocities inversely proportional to their magnitudes collide with each other, each one rebounds with the velocity that it had before the impact. WALLIS, WREN, HUYGHENS, MARIOTTE 179 " Let two bodies A and JB collide with each other. (A > B). Suppose that the velocity BC of the body B is to the velocity AC of the body A as the magnitude A is to the magnitude B. We wish to show that after the impact, A will be reflected with a velocity CA and B with a velocity CD. If the former is true for A9 the latter is true for B (conservation of relative velocity). Suppose that A is reflected with Fig. 63 a velocity CD < CA. Then B will rebound with a velocity CE > CB and DE = AB. Imagine that A has acquired its original velocity AC by falling from the height HA and that its vertical motion has then been changed into the horizontal motion with velocity A C. In the same way, suppose that B has acquired its velocity by falling from the height KB. These heights are in the square ratio of the velocities. That is, HA fCA^ 2 — -- = J — - j . Then suppose that after the impact the bodies A and B KB \CB J change their horizontal motions — whose velocities are CD and CE — into vertical upwards motions and thus arrive at the points L and M , , AL /CD\2 such that m=(m). " When the centre of gravity of A is at H and that of B at jK, their common centre of gravity is at Q. After the impact this centre of gravity is at the point N. Now it can be shown that IV is above Q. " * In this matter Huyghens invokes a principle which we find developed in some detail in the Horologium oscillatorium^ and reduces, in fact, to the principle of the conservation of living forces. 1 This is a question of pure geometry. 2 See below, p. 187. 180 THE FORMATION OF CLASSICAL MECHANICS 66 It is a well-established principle of Mechanics that, in the motion of several bodies under the influence of their centre of gravity alone, the common centre of gravity of these bodies cannot be raised. " If this principle is assumed, the supposition which has been made about the velocity with which A rebounds (CD < CA) implies a con tradiction. Huyghens dismisses the hypothesis that A is reflected with a velocity CD > CA in an analogous way. Therefore A rebounds with the velocity CA and B with the velocity CB. Q. E. D. Huyghens related all cases of the direct elastic impact of two unequal bodies to the preceding situation by using the artifice of the moving ship on every occasion. We now know that by invoking the relativity of impact phenomena, Huyghens carried the discussion into a privileged field. It follows from the rule of the composition of velocities that a percussion remains the same when a " fixed " system of reference is replaced by a " moving " one, from the moment when the relative motion of the two systems becomes continuous. With this restriction alone, the relative motion of the two systems can be accelerated in any way. Huyghens, however, restricted himself to a uniform and rectilinear relative motion — namely, that of the ship with respect to the river-bank. We add that in using the principle of inertia, on the other hand, Huyghens confined himself to an infinite number of reference points, to day termed absolutes. In fact, the principle of inertia is irrelevant to impact phenomena because of their instantaneous character. In all his writings, Huyghens took care to explain his hypotheses clearly and to deduce his propositions from them logically. His style is similar to that of Archimedes. By this means a perfect clarity is achieved at the price of some tedium. However, the rigour of this work is sometimes only an apparent one. Jouguet has come to the following conclusion after an exhaustive study of the interdependence of certain of Huyghens' hypotheses. 1 It was sufficient for Huyghens to assume the conservation of the total living force of the system in every system of reference, or its conservation in two arbitrary systems in continuous (and not zero) relative motion with respect to each other. Such a hypothesis is itself equivalent to the twin hypothesis of the conservation of living force in one arbitrary system of reference and the simultaneous conservation of the total quantity of motion in direction and sign as Wallis intended — not in Descartes' sense of absolute value. 1 L. M., Vol. I, p. 151. WALLIS, WREN, HUYGHENS, MARIOTTE 181 4. THE PLAN OF HUYGHENS' FUNDAMENTAL TREATISE. We now come to Huyghens* major work in dynamics — the treatise Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (Paris, 1673). 1 This work consists of five parts — a description of the clock; on the fall and motion of bodies on a cycloid ; the evolution and dimensions of curved lines ; on the centre of oscillation or agitation ; and finally, on the construction of a new clock with a circular pendulum, and theorems on the centrifugal force. Huyghens had constructed cycloidal clocks at The Hague since 1657. He planned this treatise over a long period of time and only completed it in 1669. In 1665 he was invited hy Colbert to visit Paris and to work at the Academy of Sciences, where he obtained a royal monopoly for the reproduction of his clocks. As early as 1667 there is mention of " three clocks made at Paris, at the expense of the King, to be used on the voyage to Madagascar. " Huyghens had already tried out his marine clock aboard an English vessel in 1664. Aware of the defect of the isochronism of the finite oscillations of a circular pendulum, Huyghens strove to find a pendulum which might be theoretically isochronous for all amplitudes. " It is the oscillations of marine clocks that most noticeably become unequal, because of the ship's continual shaking. So that it is necessary to take care that oscillations of large and small amplitudes should be isochronous. " Huyghens also made astronomical clocks, both at Leyden and Paris, that were correct to one second a day. " We have, " he wrote, " regarded the cyloid as the cause of this pro perty of isochronism that we have found, without having the least understanding of anything except that it is consistent with the rules of the craft. " In theory, " it has been necessary to corroborate and to extend the doctrine of the great Galileo on the fall of bodies. The most desired result and, so to speak, the greatest, is precisely this pro perty of the cyloid that we have discovered. ... To be able to relate this property to the use of pendulums, we have had to study a new theory of curved lines that produce others in their own evolution. " The question here is that of the theory of developable curves, and Huyghens established that the development of a cycloid is an equal cycloid. Huyghens describes his " automatic " at great length. In this clock the motion of the pendulum is determined by pulleys that are actuated 1 The complete Works of Christiaan Huyghens, published by the Dutch Society of the Sciences, The Hague, 1934, Vol. XVII. 182 THE FOBMATION OF CLASSICAL MECHANICS by the driving weights — " the oscillations of the pendulum impose the law and the rule of motion on the whole clock. " He also evolved a new and ingenious suspension which assured the continuity of the clock's motion when the driving weights were removed and replaced. Fig. 64 We return to the question of the cycloidal pendulum. The pendu lum oscillates between two thin plates whose function is to assure the constancy of the period in spite of variations in the amplitude. These plates, Km and Ki9 are cut from two half- cycloids, KM and KI. The pendulum KMP has a length equal to twice the diameter of the generating circle. Huyghens wrote, "I do not know whether any other line has this remarkable property, namely that of describing itself in its evolution " — in other words, of being identical with its development. This mathematical problem was to be taken up by Euler. 5. HUYGHENS AND THE FALL OF BODIES. Though, as we have remarked in connection with the laws of impact, Huyghens was a Cartesian to the extent that he used the relativity of motion, he was much more a direct successor of Galileo and Torri- celli, and provided a link between them and Newton. To use his words, he " corroborated and extended Galileo " in the matter of the fall of bodies. He clearly stated the principle of inertia and the principle of the composition of motions, and applied these principles to the fall of bodies and to rectilinear uniform motion in any direction. " Each of these motions can be considered separately. One does not disturb WALLIS, WREN, HUYGHENS, MARIOTTE 183 the other. " He accepted Galileo's laws on the rectilinear fall of bodies and improved the associated demonstrations. For example, he established the following proposition. " Proposition I. — In equal times the increases of the velocity of a body which starts from rest and falls vertically are always equal, and the distances travelled in equal times form , ^ a series in which successive differences are constant. M n , „ " Suppose that a body, starting from rest at A, falls through the distance AB in the first time and has acquir- ed a velocity at JB which would allow it to travel the distance BD during the second time. \£ " We know then that the distance travelled in the second time will be greater than BD, since the distance BD would be travelled even if all the action of the weight ceased at B. In fact, the body will be animated by a compound motion consisting of the uniform motion which would allow it to travel the distance BD and the motion of the fall of bodies, by means of which it must necessarily fall a distance AB. Therefore at the end of the second time, the body will arrive at the point E which is obtained by adding to BD a length DE which is equal to AB. ..." Huyghens showed in the same way that, at the end of the third time, the body will arrive at G — a point such that EF—2BD and FG=AB. The same procedure was repeated. Thus Huyghens arrived at the following proposition. Fl&- 65 " Proposition II. — The distance that a body starting from rest travels in a certain time is half the distance it would travel in uniform motion with the velocity acquired, in falling, at the end of the time considered. " To show this, Huyghens considers the distances AB, BE, EG and GK travelled in the first four intervals of time. He doubles the value of these times so that the body travels along AE in the first instant and along EK in the second. Necessarily „ BE = AD BE EK AE ~ AB ~~~ AB Now KE=2AB + 5BD and Therefore KE—EA = 4BD 184 THE FORMATION OF CLASSICAL MECHANICS 4BD AD—AB BD and consequently __ = ___=—. Therefore AE = 4>AB and BD = 2AB. Q. E. D. These demonstrations have been quoted because they differ from those of Galileo. In particular, they make use of a composition of the velocity acquired and the new fall of the body at each instant. With a little good-will it is possible to regard them as the expression of the ideas of the Schoolmen of the XlVth Century in a more sophis ticated mathematical language. Thus Buridan, in particular, believed that heaviness continually caused a new impetus to the one that was already present. Huyghens then sets out to establish a hypothesis " that Galileo asked should be granted to him as obvious. " Thus Salviati, in the Discorsi, had been obliged to take the following principle as a postulate.1 " The velocities acquired by a body in falling on differently inclined planes are equal when the heights of the planes are so. " B D Fig. 66 "... Let there be two inclined planes whose sections by a vertical plane are AB and CB. Their heights, AE and CD, are equal. I maintain that in these two circumstances the velocity acquired at B is the same. Indeed, if in falling along CB the body acquires a velocity that is smaller than in falling along AB, this velocity will be equal to that which would be acquired in some descent FB < AB. But along CB the body acquires a velocity that allows it to rise again along the whole length of BC. [This in virtue of proposition IV, which we have not quoted and which demonstrates this fact for a rectilinear rise.] Therefore it will acquire, along FB, a velocity which can make it rise again along BC — this can be achieved by reflection at an oblique surface. It will therefore rise as far as C, or to a height greater than that from which it fell, which is absurd. " 1 See above, p. 135. WALLIS, WREN, HUYGHENS, MARJOTTE 185 In the same way it is shown that in descending along AB the body cannot acquire a smaller velocity than in falling along CB. This establishes the proposition. Huyghens shows that the durations of fall have the same relation to each other as the lengths of the planes. He also shows that when the body falls in a continuous motion from a given height along any number of differently inclined planes, it always acquires the same velocity as that obtaining at the end of a vertical fall from the same height. Conversely, in rising again along a trajectory formed of contiguous and differently inclined planes, the body will achieve its original height (Proposition IX). A passage to the limit then allows the question of the motion of a body on a curve contained in a vertical plane to be considered. 6. THE ISOCHRONISM OF THE CYCLOIDAL PENDULUM. Huyghens arrived at a proof of the isochronism of a cycloidal pendulum by means of an argument in infinitesimal geometry. Ad mirable though this was, it required no less than a dozen propositions, and we cannot reproduce it here.1 The principal result is stated in the following terms. " Proposition XXV. — In a cycloid whose axis is vertical and whose summit is placed below, the times of descent in which a particle starts from any point on the curve and reaches the lowest point are equal to each other ; their ratio with the time of vertical fall along the whole axis of the cycloid is equal to the ratio of half the circumference of a circle to its diameter. " This result may be obtained easily by means of a well-known analysis. Thus the motion of a heavy particle on a cycloid is defined by the differential equation dt2 4R Here 5 denotes the distance of the particle from A as measured along the arc. If the particle starts from rest at the point J3, and rr if the arc AB is equal to $0, it follows that s = s0 cos y ~- - 1. The sum- jt /4iR mit, A, is attained in a time T= — V/ — • Now if the particle were o 1 Cf. The Complete Works of Christiaan Huyghens, Vol. XVIII, pp. 152 to 184. 186 THE FORMATION OF CLASSICAL MECHANICS allowed to fall freely along DA, it would reach A after_a time T given by 2R == i gT'2, from which it foUows that T = V/ — • Therefore it <£ O must be that — = -, which is the statement that Huyghens makes.1 T 2 B Fig. 67 We shall pass over the third part of Huyghens' treatise, which is devoted to curved lines. The question is that of the search for developments. Huyghens called the development of a curve, the evoluta, and the development the descripta ex evolutions. Notable among this work is the development of the cycloid, which rationally justified the use of the cycloidal shape in his clock. He also studied the developments of conies — in particular, those of the parabola, which he called paraboloides.2 7. THE THEORY OF THE CENTRE OF OSCILLATION. "We now come to the fourth part of the Horologium oscillatorium, which is devoted to an investigation of the centre of oscillation. " A long time ago, when I was still almost a child, the very wise Mersenne suggested to me, and to many others, the investigation of centres of oscillation or agitation. " Thus Huyghens expresses him self at the beginning of the fourth part of his major work. At first he found nothing "which might open the way to this contemplation. " However, he returned to the question in order to improve the pendulums of his " automatic, " to which he had been led to add moveable weights above the principal fixed weight. Huyghens completely resolved 1 It is of some interest to remark that this study of the cycloid was very popular among XVIIth Century geometers. WHEN had calculated its length, ROBERVAL had defined the tangents while PASCAL determined the centre of gravity and calculated the area. WALLIS too, had made analogous investigations. 2 HUYGHENS knew the development of a parabola as early as 1659, as a result of his work with Jean VAN HEURAET of Harlem. WALLIS, WREN, HUYGHENS, MARIOTTE 187 this question by appealing to a kind of generalisation of TorricellTs principle that depended on the principle of living forces. First he defines the compound pendulum and the centre of oscillation. The latter is the point on the perpendicular to the axis of oscillation through the centre of gravity which is separated from the axis by a distance equal to the length of the simple isochronous pendulum. Huyghens starts from the following fundamental hypothesis. " We suppose that when any number of weights starts to fall, the common centre of gravity cannot rise to a height greater than that from which it starts. " In the commentary which accompanies this hypothesis, Huyghens specifies that verticals should be considered as parallels if the conside ration of a centre of heaviness is to have any meaning. His hypothesis reduces to the following — no heavy body can rise by the sole agency of its own gravity ; what is true for a single body is also true for bodies which are attached to each other by rigid rods. If, now, the bodies considered are no longer connected to each other, they nevertheless have a common centre of gravity, and it is this which cannot rise spontaneously. 188 THE FORMATION OF CLASSICAL MECHANICS " Let there be weights A, B, C and let D be their common centre of gravity. Suppose the horizontal plane is drawn and that EDF is a right section of it. Let DA, DB, DC be the rigid lines joining the points to each other in a rigid way. Now set the weights in motion so that A comes to E in the plane EF. Since all the rods are turned through the same angle, B will now be at G and C at H. " Finally suppose that B and C are joined by the rod HG which cuts the plane EF in F. The point F must also be the centre of gravity of these two weights taken together, since D is the centre of gravity of the three weights at E, G, H and that of the body E is also in the plane DEF. The weights H and G are once more set in motion about the point F as about an axis and, without any force, simultaneously arrive in the plane EF. Thus it appears that the three weights, which were originally at A, B and C, have been carried exactly to the height of their centre of gravity by their own equilibrium, Q. E. D. The demonstra tion is the same for any other number of weights. " Now the hypothesis that we have made is also applicable to liquid bodies. By its means, not only may all that Archimedes has said about floating bodies be demonstrated, but also many other theorems in mechanics. And truly, if the inventors of new machines who strive in vain to obtain perpetual motion were able to make use of this hypothesis, they would easily discover their errors for themselves and would under stand that this motion cannot be obtained by any mechanical means. " Huyghens' second hypothesis consists of the neglect of the resistance of the air and all other disturbances of motion. His first three propositions relate to the geometry of masses. We come to the fourth. " Proposition IV. — If a pendulum composed of several weights, and starting from rest, has executed some part of its whole oscillation, and it is imagined that, from that moment on, the common bond of the weights is broken and that each of the weights directs its acquired velocity upwards and rises to the greatest height possible, then by this means the common centre of gravity will rise to the height it had at the start of the oscillation. " Let a pendulum composed of any number of weights A, B, C be connected by a weightless rod which is suspended from an axis D per pendicular to the plane of the diagram. The centre of gravity, E, of the weights A, B, C is supposed to be in this plane. The line of the centre, DE, makes an angle EDF with DF and the pendulum is drawn aside as far as this. Suppose that it is released in this position and that it executes a part of its oscillation in such a way that the weights A, B, C come to G, H, K. Suppose that each of these weights directs its velocity upwards when the bond is broken (this can be arranged by WALLIS, WREN, HUYGHENS, MARIOTTE 189 the adjunction of certain inclined planes) and rises to the greatest possible height, as far as L, M, N. Let P be the centre of gravity of the weights when they have attained these positions. I maintain that this point is at the same height as E. R B Fig. 69 " First, it is certain that P is not higher than E (hypothesis I). But neither is it at a lesser height. Indeed, if this is possible, let P be lower than E. Suppose that the weights fall down p™*™^*? ™ heights that they travelled in mounting— namely LG, Mti, I\^. it clear that they will attain the same velocity as they had at the beginning of their climb-that is to say the velocity they acquired m the motion of the pendulum from CBAD to KHGD. Consequently ^ jhey are simultaneously attached again to the rod which supported ^em, they will continue their motion along the arcs winch they had started along (This will happen if, before coming to the rod they rebound on the planes 00.) The pendulum reconstituted in this way will effect the rest of its motion without any interruption. So that the centre of gra vity E travels, in rising and falling, along the equal arcs EF and *«, and finds itself at R-at the same height as at E. But we have supposed that R is higher than P, the centre of gravity of the weights whe* .they are at L, M, N. Therefore R will also be higher than P. The centre of gravity of the weights which have fallen from L M, N will herefore have risen by a height greater than that from which they fell which is absurd. The centre of gravity P is not, therefore, lower than E. 190 THE FORMATION OF CLASSICAL MECHANICS No more is it at a greater height. It must therefore be that it is at the same height. Q. E. D. " " Proposition V. — Being given a pendulum composed of any number of weights, if each of these is multiplied by the square of the distance from the axis of oscillation, and the sum of these products is divided by the product of the sum of the weights with the distance of their centre of gravity from the same axis of oscillation, there will be obtained the length of the simple pendulum which is isochronous with the compound pendulum — that is, the distance between the axis and the centre of oscillation of the com pound pendulum. " We shall analyse this demonstration instead of reproducing it. Let A, JB, C be the material points which constitute the compound and a, ft, c be their weights. Suppose that DA = e, DB = /, DC = g and ED = d. Also suppose that E is the centre of gravity of the weights. Initially the compound pendulum is released from rest in the position DABC. Let FG be a simple pendulum isochronous with the compound pendulum and placed, initially, at FG. Let the angle FGH be equal to the angle EDF. On DE, mark oS the length of the simple pendulum, Fig. 70 x — FG. There is isochronism between the simple pendulum and the compound pendulum if, at corresponding points of the two oscillations, 0 and P — such that the arc GO is equal to the arc LP — the velocities of G and L are equal. WALLIS, WREN, HUYGHENS, MARIOTTE F 191 We shall show that this equality holds for - (a + b + C)d Indeed, suppose that the velocity from L to P is greater than that from G to 0. Let SP, RQ, YO be the descents from the points L, JE, G to the corresponding points P, (), 0. d If SP = y, then RQ = y -. re The simple pendulum G has a velocity at 0 which is sufficient to enable it to return to the height of M, either along OM or along OY by means of a suitably chosen elastic impact. Then the point L has, at P, a velocity greater than that which would enable it to return along PS = OY. Let hL be greater than y, the height to which L can return. The points A9 JS, C travel the arcs AT, BF, CX while L travels LP. Thus for yi there obtains the relation v(A) to T D.A ^ e_ v(L) to P "" DL ~~~ x Now the heights of return are proportional to the squares of the velocities. Therefore the height of return of A9 say ft^, is greater than e2 — y, from the moment that L exceeds y. 192 THE FORMATION OF CLASSICAL MECHANICS The same is true for B and C, so that Now this inequality expresses the fact that the centre of gravity E can return to a greater height than the one — RQ — from which it fell, for hE = . This result is in contradiction with pro- a + b + c r position IV and therefore impossible. In the same way the hypothesis that the velocity from L to P will be less than the velocity from G to 0 implies a contradiction with the same proposition. Therefore the pen dulum FG of length x is synchronous with the compound pendulum. This establishes the required result. We shall not discuss the applications of these propositions here, but shall indicate how Huyghens was able to demonstrate the reciprocity between the axis of suspension and the axis of oscillation. Huyghens stated the following proposition. " Proposition XVIII. — If the plane space, whose product with the number of particles of the suspended magnitude is equal to the sum of the squares of their distances from the axis of gravity., is divided by the distance between the two axes, the result obtained is the distance from the centre of gravity to the centre of oscillation. " In this enunciation, the axis of gravity is the axis through the centre of gravity and parallel to the axis of suspension. yya Huyghens' plane space has the value •= — , where r' is the distance n from the axis of gravity of one of the n equal particles constituting the suspended magnitude. It is therefore identical with the square of the radius of gyration of the pendulum, £2, about this axis. If x denotes the length of the simple isochronous pendulum and d the distance between the centre of gravity and the axis of suspension, Huyghens' statement may be written nd a V 2 Now, because of proposition V, the length x is equal to , where nd r is the distance from one of the n particles to the axis of suspension. This is equal to — , where K is the radius of gyration of the pendulum WALLIS, WREN, HUYGHENS, MARIOTTE 193 ibout this axis. Huyghens' long demonstration of proposition XVIII •educes, then, to the verification of the equality This is an immediate consequence of the very definition of the mo ment of inertia, which was to he introduced by Euler. Huyghens then states the following proposition. 66 Proposition XIX. — When the same magnitude oscillates, the suspen sion being sometimes shorter, sometimes longer, the distances from the centre of oscillation to the centre of gravity are inversely proportional to the distances from the axes of suspension to the centre of gravity. " r™ • . . i -i . x — d dr ., . Ihis statement is equivalent to the equation —f - - = — and is a xf — d' d direct consequence of proposition XVIII. Finally, Huyghens was able to state the reciprocity of the two axes, " Proposition XX. — The centre of oscillation and the point of sus pension are reciprocal. " This reciprocity is a direct consequence of Proposition XIX and the constancy of the product d (x — d). In Huyghens' work the whole theory of the centre of oscillation rests on the fundamental hypothesis described on page 187. This is equi valent to the a priori assumption of the conservation of living forces. In Lagrange's opinion, he thus sets out from an " indirect precept. " Huyghens' theory produced its critics, like Roberval, Catelan, Jacques Bernoulli and others — but these had nothing more than " evil objec tions. " x However, criticism had the value of attracting the attention of geometers — as did the efforts of Descartes and Roberval to solve the same problem — to the investigation of the velocities lost or gained in the constrained motion of the elementary weights that constitute a com pound pendulum. Jacques Bernoulli was at first mistaken in this investigation, in that he considered the velocities acquired in a finite time. The Marquis de 1'Hospital drew his attention to the fact that an infinitesimal motion of the system should be considered. It was due to this remark, made in 1680, that Jacques Bernoulli arrived at a new solution of the problem of the centre of oscillation (1703). We shall return to a discussion of this solution, which prepared the way for d'Alembert's principle. 1 Mfaanique analytique, Part II, Section I. 13 194 THE FORMATION OP CLASSICAL MECHANICS 8. THE THEORY OF CENTRIFUGAL FORCE. The Horologium oscillatorium finishes with thirteen unproved propo sitions on centrifugal force and the conical pendulum. These proposi tions were the subject of De vi centrifuga. This manuscript was written in 1659 but did not appear until 1703, in a form that was edited by de Voider and Fullenius and published posthumously with the remainder of Huyghens work. In this treatise Huyghens regards gravity as a tendency (conatus) to fall. This tendency is made apparent by the tension of the thread which supports a body. To measure it, it is necessary to consider the first motion of the body after the thread has been broken. In this way the conatus is caught in life, before there has been time for it to have been destroyed. Given this, Huyghens sets out to determine the conatus of a body attached to a revolving wheel. By an artifice whose object is clearly that of introducing a reference system bound to the wheel, he assumes that the wheel is sufficiently large to carry a man who is attached to it. This man holds a thread, supporting a ball of lead, in his hand. Because of the rotation the thread is stretched with the same force as if it were fixed at the centre of the wheel. In equal times the man travels the 0 6 Fig. 72 very small arcs BE and B F. If it is released at B, the lead will travel along the rectilinear paths BC and CD which are equal to these arcs. The points C and D do not fall on the radii AE and EF, but very slightly behind them. WALLIS, WREN, HUYGHENS, MARIOTTE 195 If the points C and D coincide with y and <?, points on the radii AE, 4F, the lead will tend to move away from the man along the radius. The distances JEJy, F<5, . . . increase as the square numbers 1, 4, 9, 16, . . . and this becomes more accurate as the arcs BE, jEF, . . . become smaller. Now, according to Galileo's laws, the distances travelled by a body that starts its fall from rest increase as the successive square numbers 1, 4, 9, 16, ... The conatus which is sought will therefore be, on this hypothesis, the same as that of a heavy body suspended by a thread. In fact, however, the points C and D lie behind y and d. Therefore, with respect to the radius on which it is placed, the weight tends to describe a path which is tangential to the radius. But at the moment of the separation of the lead and the wheel, these curves can be regarded as being the same as their tangents J5y, F<3, . . . with the consequence that the distances EC, FD, . . . must be considered as increasing as the series 1, 4, 9, 16, . . . And here is Huyghens' conclusion. " The conatus of a sphere attached to a revolving wheel is the same as if the sphere tended to advance along the radius with a uni formly accelerated motion. ... It is sufficient, indeed, that this motion should be observed at the beginning. Afterwards, the motion can follow every other law. This cannot affect the conatus that exists at the beginning of the motion in any way. This conatus is entirely similar to that of a body hung by a thread. From which we conclude that the centrifugal forces of unequal particles that move with equal velocities on equal circles have the same relation to each other as their gravities, that is, as their quantities of solid (quantitates solidae) . [Here we catch a fleeting glimpse of the concept of mass.] Indeed, all bodies tend to fall with the same velocity in the same uniformly accelerated motion. But their conatus has a moment (momentum) that is greater as the bodies themselves are greater. It must be the same for bodies that tend to move away from a centre, since their conatus is similar to that which arises from their gravity. But while the same sphere always has the same tendency to fall whenever it is hung from a thread, the conatus of a sphere attached to a revolving wheel depends on the velocity of rotation of the wheel. It remains to us to find the magnitude or the quantity of conatus for different velocities of the wheel. " l So much for the principle of centrifugal force that Huyghens developed in his preamble. We shall now state the propositions that he established in a much abbreviated form. 1) For a given period of rotation, the centrifugal force is propor tional to the diameter. 1 The Complete Works of Christiaan Huyghens, Vol. XVI, p. 266. 196 THE FORMATION OF CLASSICAL MECHANICS 2) For a given velocity on the circumference, it is inversely propor tional to the diameter. 3) For a given radius, it is proportional to the square of the velocity on the circumference. 4) For a given centrifugal force, the period of revolution is propor tional to the square root of the radius. 5) " When a particle moves on the circumference with the velocity that it would have acquired in falling from a height equal to a quarter of the diameter, its centrifugal force is equal to its gravity. In other words, it will stretch the cord to which it is attached with the same force as if it were suspended. " We shall summarise Huyghens' proof of this proposition. The particle is supposed to describe the circumference of a circle with uniform motion and the velocity (\/Kg) which it would have T) acquired in falling from the height CJB = — . If it is detached at B, it 2i travels along the tangent uniformly, covering a distance BD = R in the time I i/ __ J which it would have spent in falling along CB. We consider \ o / a very small fraction of BD — namely BE — and draw the straight line C C* / R T?\ ^ EFAH . We also suppose that _==(_). Then BE, or ( <\/2R.CG) , Ojt> \ JoXx / is proportional to the time of free fall along CG, which is equal to V1- 8 WALLIS, WREN, HUYGHENS, MARIOTTE 197 The particle detached at B travels the distance EE uotformly in the time it would have spent in falling freely from the height CG. Now BE can be approximated to by the arc BF. If it is shown that CG^FE it will have been proved that the conatus of the centrifugal force is equal to the conatus of the gravity, for the particle considered. - B£2 = — i—\\= CG. The proposition is etablished. — 2R 2 \ JR / *• It must be pointed out that for Huyghens centrifugal force is in no way a fictitious force. On the contrary, he attributed to it both measure ment and special privilege by identifying it with gravity in the part- icular case we have just seen. Let us continue our examina tion of the propositions that end the Horologium oscillatorium : 6) A conical pendulum is iso chronous with a simple pendulum having as length half the latus rectum (parameter) of the par aboloid of which it describes a parallel. 7) The period of a conical pen dulum depends only on the height of the cone. 8) It is proportional to the square root of this height. 9) The period of motion of a conical pendulum " on extremely small circumferences " is equal to n T, T being the time it falls freely from a height twice the length of the pendulum. 10) If a mobile runs along a circumference and if the period of its uniform motion is equal to the time in which a conical pendulum, whose length is equivalent to the • «. •* i radius, describes an extremely small circumference, its centrifugal force is equal to its gravity. . 11) The period of revolution of a conical pendulum is equal to the time it takes to fall freely from a height equal to its length, if the string Fig. 74 198 THE FORMATION OF CLASSICAL MECHANICS forms with the horizontal plane an angle the sine of which is equal to 12) The tension of the string of a conical pendulum of given height is proportional to the length of the pendulum. 13) When a simple pendulum performs a maximum lateral motion, that is, when it descends by a whole quarter of the circumference, the tension of the string at the lowest point is three times the weight of the pendulum.1 Without spending time over the demonstrations of these propositions, we may mention, for the sake of curiosity, a clock constructed by Huyghens, which illustrates this theory. The axis KH is vertical, the curved line A I is the e volute of the parabola FEC. In the rotation of the axis, the pendulum BF, which escapes tangentially from the evolute, describes a parallel, a straight section of the paraboloid engendered by FEC. 9. HUYGHENS AND THE PRINCIPLE OF RELATIVITY. In Huyghens' sense, the principle of relativity is that it is impossible that an observer in uniform rectilinear motion can discover his own translation. We have seen how Huyghens exploited this principle in his study of the laws of impact. Huyghens appears to have assumed, however, in his work on the centrifugal force, the tangible reality of uniform, circular motion, which he called motus verus. Nevertheless, he went back on this opinion after the appearence of Newton's Principia. In the last analysis, he rejected the concept of absolute motion and remained a Cartesian. Indeed, in a fragment of his writing that must be placed later than 1688, he wrote, " In circular motion as well as in straight and free motion there is nothing that is not relative, in the sense that this is all there is to know of motion. " We do not have the time to deal with Huyghens' contribution to physics. We only recall that he outlined an undulatory theory of light in 1673 — he had learnt of Erasme Bartholin's experimental discovery of the birefringence of Iceland Spar in 1670, and sought to find a rational explanation of this phenomenon. Huyghens read his Treatise on Light before the Academic des Sciences at Paris in 1690. 1 (Euvres completes de Huyghens, vol. XVI, p. 280 sq. WALLIS, WREN, HUYGHENS, MARIOTTE 199 10. MARIOTTE AND THE LAWS OF IMPACT. Mariotte dealt with the theory of impact in his Treatise on the percussion or impact of bodies. This work added nothing essentially novel to the work of Wallis, Wren and Huyghens, but was distinguished by its much more expe rimental approach to the subject. Mariotte rejected the concept of perfectly hard bodies in the sense that "Wallis had used the term. He confined himself to bodies that were flexible and resilient (that is, perfectly elastic) and bodies that were flexible and not resilient (that is, perfectly soft). In order to obtain velocities of direct impact that were in any given relationship, Mariotte described an apparatus consisting of two equal pendulums which were allowed to fall from positions J3J', EL' that could be chosen at will. Mariotte had the merit of recognis ing the part played by the mass in the laws of impact. Thus he wrote — " The weight of a body is not understood, here, to be the tendency which makes it move towards the centre of the Earth, but rather to be its volume together with a certain solidity or condensation of the parts of its material which is probably the cause of its heaviness. " Mariotte was also concerned with the investigation of centres of percussion. He combined the laws of statics with those of impact in this investigation. He investigated the percussions exerted on a balance by jets of water of known amount and came to the following conclusion. " Two bodies that fall on a balance, on one side and the other, are in equilibrium at the moment of impact if the distances [from the centre] of the points where they fall are in reciprocal proportion to their quantities of motion. " Fig. 75 CHAPTER SIX NEWTON (1642-1727) 1. THE NEWTONIAN METHOD. Thanks to Galileo and Huyghens, mechanics had been emancipat ed from the scholastic discipline. Essential problems like the motion of projectiles in the vacuum and the oscillations of a compound pendulum had been solved. Nevertheless, the task of constructing an organised corpus of principles in dynamics remained. This was the work of Newton, who set his seal on the foundation of classical mechanics at the same time that he extended its field of application to celestial phenomena. Newton's work in mechanics is called Philosophiae naturalis prin- cipia mathematica (1687) -1 It proceeds by a method that is at once rational and experimental, to which the author himself gave us the key. A first rule of the newtonian method consists in not assuming other causes than those which are necessary to explain the phenomena. A second is to relate as completely as possible analogous effects to the same cause. A third, to extend to all bodies the properties which are associated with those on which it is possible to make experiments. A fourth, to consider every proposition obtained by induction from observed phenomena to be valid until a new phenomenon occurs and contradicts the proposition or limits its validity. It was by relying on the third of these rules that Newton was able to formulate the law of universal gravitation. In expressing this law, Newton had no intention of assigning a cause to gravitation. " But hitherto I have not been able to discover the cause of those pro perties of gravity from phenomena, and I frame no hypotheses (hypo- 1 The manuscript of the Principia was deposited with the Royal Society on April 28th, 1686. It was published for the first time in 1687, on the intervention of HALLEY. NEWTON 201 theses non fingo). For whatever is not deduced from the phenomena, is to be called an hypothesis ; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propo sitions are inferred from the phenomena, and afterwards rendered general by induction. " 1 It is not surprising that Newton himself should have departed from this rule and that he should have introduced purely abstract entities into some of his arguments. But on the whole, his work is a practical expression of the natural philosophy whose foundation he laid. 2. THE NEWTONIAN CONCEPTS. Newton introduced the notion of mass into mechanics. This notion had appeared in Huyghen's work, but only in an impermanent form.2 " Definition /. — The Quantity of Matter is the measure of the same, arising from its density and bulk conjunctly. "... I have no regard in this place to a medium, if such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter every where under the name of Body or Mass. And the same is known by the weight of each body — For it is proportional to the weight, as I have found by experiments on pendulums, very accurately made. ..." In these experiments Newton worked with pendulums made of different materials and of the same length, and established that their acceleration did not depend on the nature of the material. He eliminat ed the variations of the resistance of the air by using pendulums formed of spheres of the same diameter, suitably hollowed-out to ensure equality of the weights. When Newton declared that the mass was known by the weight, he contemplated the weight in a given place. For he was well aware of the fact that the weight of a body varied with its distance from the centre of the Earth, while its mass remained constant. This Newtonian definition of the mass has been often and justly criticised. Thus Mach wrote, " The vicious circle is clear, since the density can only be defined as the mass of unit volume. Newton clearly believed that to each body was associated a characteristic determinant 1 In the English translation of the present book quotations from the work of NEWTON are taken from Andrew MOTTE'S translation of the Principia (1724). 2 See above, p. 195. 202 THE FORMATION OF CLASSICAL MECHANICS of its motion, which was different from its weight and which we, with him, call mass. But he did not succeed in expressing this idea correctly." l " Definition II. — The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter conjunctly. " Definition III. — The vis insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line. " To Newton, this vis insita was always proportional to the quantity of matter. He also gave it — with a meaning different from that which is accepted now — the name of force of inertia. This force is resistive when it is desired to change a body's state of motion, and impulsive to the extent that a body in motion acts on an obstacle. " Definition IV. — An impressed force (vis impressa) is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line. " Therefore, to Newton, the vis impressa is the determinant of the accel eration. " This force consists in the action only ; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis insita only. " The impressed force acts by impact, pressure or at a distance. " Definition V. — A Centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre. " As examples of centripetal force, Newton cites the gravity which makes bodies tend towards the centre of the Earth, the magnetic force that attracts iron towards a magnet, and that force — whatever its nature might be — that makes each planet describe a curved orbit. The force exerted by a hand that whirls a stone in a sling is also a centripetal force. Newton adds, " And the same thing is to be under stood of all bodies, revolved in any orbits ; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call Centripetal, would fly off in right lines, with an uniform motion. " Newton then distinguishes the absolute quantity, the accelerative quantity and the motive quantity of the centripetal force (Definitions VI, VII and VIII). The absolute quantity depends on the efficacy of the cause that propagates the centripetal force — for example, the size of a stone or the strength of a magnet. The accelerative quantity is measured by the velocity produced in 1 M., p. 190. NEWTON 203 a given time. Therefore, in modern language, it is the acceleration produced by the force. Newton takes the value of the motive quantity to be the quantity of motion produced in a given time. Therefore it is the motive quantity which satisfies the law that is now written — (1) F = my. For heavy bodies, the motive quantity becomes identified with the weight. In this way Newton multiplied the definitions and concepts. Instead of deducing the concept of motive force from the concepts of mass and acceleration by using the law (1), he consciously regarded the mass and the force as two primarily distinct notions. Newton also took certain precautions in order to anticipate the objections of the cartesian philosophy, and to make the notion of action at a distance acceptable. " I likewise call Attractions and Impulses, in the same sense, Accelerative and Motive ; and use the words Attrac tion, Impulse or Propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not Physi cally but Mathematically — Wherefore the reader is not to imagine, that by those words I any where take upon me to define the kind, or the manner of any Action, the causes and the physical reason thereof, or that I attribute Forces, in a true and Physical sense, to certain centres which are only mathematical points. " Newton then proceeds to discuss the currently used concepts of time, space, place and motion. He introduces a distinction between the relative, apparent and common senses of the words and the absolute, true and mathematical senses. " I, Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to any thing external and by another name is called duration — relative, apparent and common time is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year. 66 II. Absolute space, in its own nature, without regard to anything external, remains always familar and immoveable. Relative space is some moveable dimension or measure of the absolute spaces ; which our senses determine, by its position to bodies ; and which is vulgarly taken for immoveable space. ..." Or again, " It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may 204 THE FORMATION OF CLASSICAL MECHANICS be accelerated and retarded, but the true, or equable progress, of abso lute time is liable to no change. . . . " For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space, as to order of situation. It is from their essence or nature that they are places ; and translations out of those places, are the only absolute motions. " Newton concerns himself with distinguishing absolute and relative motions by their causes and their effects. " The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed on the body moved — but the relative motion may be generated or altered without any force impressed upon the body. . . . " The effects which distinguish absolute from relative motion, are the force of receding from the axe of circular motion. For there are no such forces in a circular motion, purely relative, but in a true and absolute circular motion, they are greater or less, according to the quan tity of motion. " If a vessel, hung by a long cord, is so turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move — but the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced) and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axe of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest it produced no endeavour to recede from the axe — the water shewed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel, proved its endeavour to recede from the axe ; and this endeavour shewed the real circular motion of the water NEWTON 205 perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. ..." Again Newton stresses the distinction between absolute and relative quantities. " And if the meaning of the words is to be determined by their use ; then by the names time, space, place and motion, their measures are properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon this account, they do strain the Sacred Writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of Mathematical and Philosophical truths, who confound real quantities themselves with their relations and vulgar measures, " Newton did not conceal the difficulty of distinguishing true from apparent motions, because " the parts of that immoveable space in which those motions are performed, do by no means come under the observation of our senses. " In order to accomplish this it is necessary, according to him, to make use simultaneously of the apparent motions, u which are the differences of the true motions " and the forces, " which are the causes and the effects of the true motions. " As an example he cites the motion of two spheres attached by an inflexible thread and turning about their centre of gravity. The tension of the thread allows " the quantity of circular motion " to be measured. To Newton, force therefore appears as a true or absolute element and is opposed to motion, which only has a relative character with respect to a suitably chosen reference system. Certain modern critics — notably Mach — reproach the Newtonian philosophy for its metaphysical cha racter in this connection. Absolute space and time appear to them as purely abstract entities which can only be deduced from observation. More correctly, theoretical physics is based on the introduction of pure unobservables as intermediaries in the calculation. Under a cloak of metaphysical appearance it contains a profound physical truth. It " explicitly proclaims to the student of mechanics the necessity of considering the privileged reference frames in time and space and of thus avoiding the confusion that is so apparent in the ideas of Descartes and Huyghens."1 3. THE NEWTONIAN LAWS OF MOTION. Newton stated the principle of inertia at the beginning. This had already been discovered by Galileo and reformulated by Huyghens. " Law L — Every body perseveres in its state of rest, or of uniform 1 JOUGUET, L. M., Vol. II, p. 11, note 9. 206 THE FORMATION OF CLASSICAL MECHANICS motion in a right line, unless it is compelled to change that state by forces impressed thereon. " He next repeats the idea that the motive force is the determinant of acceleration. " Law II. — The alteration of [the quantity of] motion is ever propor tional to the motive force impressed ; and is made in the direction of the right line in which that force is impressed, " The third law constitutes the principle of the equality of the action and the reaction. " Law III. — To every action there is always opposed an equal reaction — or the mutual actions of the two bodies upon each other are always equal, and directed to contrary parts. " Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tyed to a rope, the horse (if I may so say) will be equally drawn back towards the stone — For the distended rope, by the same endeavour to relax or unbend it self, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. " If a body impinge on another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. " The changes made by these actions are equal, not in the velocities, but in the [quantities of] motion of the bodies ; that is to say, if the bodies are not hindered by any other impediments. For because the motions are equally changed, the changes of velocity made towards contrary parts, are reciprocally proportional to the bodies. This law also takes place in attractions. ..." It is interesting to see how Newton — contrary to the custom of the time — pays homage to his predecessors. " Hitherto I have laid down such principles as have been received by all mathematicians, and are confirmed by abundance of experiments. By the two first Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola ; expe rience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air. . . . " On these same laws and corollaries depend those things which have been demonstrated concerning the times of vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By NEWTON 207 the same together with the third Law Sir Christopher Wren, Dr. Wallis and Mr. Huyghens, the greatest geometers of our times, did severally determine the rules of the congress and reflection of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves, as to those rules, Dr. Wallis indeed was something more early in the publication ; then followed Sir Christopher Wren, and lastly, Mr. Huyghens. But Sir Christopher Wren confirmed the truth of the things before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. " For his part, Newton repeated the experiments with great care. From them he concluded that " the quantity of motion, collected from the sum of the motions directed towards the same way, or from the difference of those that were directed towards contrary ways, was never changed, " whether the bodies were hard or soft, elastic or not. In order to justify the equality of action and reaction in the case of attractions, Newton argued in the following way. " Suppose an obstacle is interposed to hinder the congress of any two bodies A, B, mutually attracting one the other — then if either body as A, is more attracted towards the other body jB, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B ; and therefore will not remain in aequilibrio — but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly towards the parts on which B lies ; and in free spaces, to go forward in infinitum with a motion perpetually accelerated. Which is absurd, and contrary to the first law. For by the first law, the system ought to persevere in it's state of rest, or of moving uniformly forward in a right line ; and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. " I made the experiment on the loadstone and iron. If these placed apart in proper vessels, are made to float by one another in standing water ; neither of them will propel the other, but by being equally attracted, they will sustain each others pressure, and rest at last in equilibrium. " 4. NEWTON AND THE DYNAMICAL LAW OF COMPOSITION OF FORCES. We have seen how Stevin and Roberval had established the rule of the composition of forces in statics. Newton arrived at the law of the parallelogram of forces by purely dynamical considerations. 208 THE FORMATION OF CLASSICAL MECHANICS " Corollary I (to the second law). — A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart. " If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion be carried from A to B ; and by the force N impressed apart in the same place, should be carried from A to C — compleat the parallelogram 76 ABCD, and by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the force N be impressed or not ; and therefore at the end of that time, it will be found somewhere in the line jBD. By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D by Law I. " Newton's demonstration is clearly based on the postulate of the independence of forces. The words " with an uniform motion " show that he considered the impulsion of the force M or N during an infinitely short time. This force acts instantaneously, like a percussion — this explains the importance of the laws of impact in Newton's thought. The students of mechanics in the XVIIth Century had all perceived that the phenomenon of impact was a means of crystallising the effect of a force into the velocity acquired in a first instant. " Corollary II. — And hence is explained the composition of any one direct force AD, out of any two oblique forces AB and BD ; and, on the contrary the resolution of any one direct force AD into two oblique forces AB and BD — which composition and resolution are abundantly confirmed from Mechanics. " Newton deduces the condition of equilibrium for simple machines (the balance, inclined plane and wedge) from this proposition. We have already seen that Aristotle compounded motions according to the rule of the parallelogram.1 Since the force was the determinant of the velocity in his belief, it may be held, as Duhem has done, that Aristotle compounded forces in the same way. 1 See above, p. 21. NEWTON 209 For Newton too, the composition of forces according to the rule of the parallelogram had an origin in dynamics. But, to him, the force was the generator of a quantity of motion in a given elementary time (Definition VIII, para. 2 above). 5. THE MOTION OF A POINT UNDER THE ACTION OF A CENTRAL FORCE. By a very simple and direct geometrical argument, Newton esta blished that the motion of a material point that is subject to a central force was contained in a plane, and followed the law of areas which Kepler had formulated in a semi-empirical way (the radius vector sweeps through equal areas in equal times). Here is Newton's argument. " Suppose the time to be divided into equal parts, and in the first part of that time, let the body by its innate force describe the right line AB. In the second part of that time, the same would (by Law I), if not hindered, proceed directly to c, along the line Be equal to AB ; so that by the radii AS, BS, cS drawn to the centre, the equal areas A SB, J3Sc, would be de scribed. But when the body arriv ed at B, suppose that a centripetal force acts at once with a great im pulse, and turning aside the body from the right line Be, compells it afterwards to continue its motion along the right line J5C. Draw cC parallel to BS meeting JBC in C ; and at the second part of the time, the body (by Cor. I of the laws) will be found in C, in the same plane with the triangle ASB* Join SC, and, because SB and Cc are parallel, the triangle SBC will be equal to the triangle SJ3c, and therefore also to the triangle SAB. " By the like argument, if the centripetal force acts successively in C, D, E, and c and makes the body in each single particle of time, to describe the right lines CD, DE, EF, and c they will all lye in the same plane ; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore in equal times, equal areas are described in one immoveable plane — and, by composition, any sums SADS, SAFS, of those areas, are one to the other, as the times 1 We shall encounter this method of argument again in the next paragraph. It is equivalent to making use of the deviation (in the kinematic sense) produced by the force during an infinitely short time. 14 210 THE FORMATION OF CLASSICAL MECHANICS in which they are described. Now let the number of those triangles be augmented, and their breadth diminished in infinitum ; and their ulti mate perimeter ADF will be a curve line " Newton establishes the converse of this proposition. He then exa mines the circular trajectory of a body gravitating about the centre of this trajectory — its gravity is equal to the centripetal force. Therefore the gravity can be evaluated by using the propositions given by Huyghens in his Horologium oscittatorium.1 Newton then studies a particle which describes a circular orbit under the action of a force emanating from any point in the plane of the circle. Given this, he comes to the fundamental problem of the motion of the planets. 6. NEWTON'S EXPLANATION OF THE MOTION OF THE PLANETS. We shall quote the original text of the Principia (De motu corporum. Liber I9 Prop. VI, cor. 5). " Si corpus P revolvendo circa centrum S describat lineam curvam APQ ; tangat vero recta ZPR curvam illam in puncto quovis P et ad Fig. 78 tangentem ab alio quovis curvae puncto Q agatur QR distantiae SPparallela, ac demitatur QT perpendicularis ad distantiam illam SP : vis centripeta SPquad X QT quad ent reciproce ut sohdum — si modo solidi illius ea QR semper sumatur quantitas quae ultimo sit., ubi coeunt puncta P et Q. 44 Nam QR aequalis est sagittae dupli arcus QP in cujus media est P ; et duplum trianguli SPQ, sive SP X QT, tempori quo arcus iste duplus describitur^proportionale est ; ideoquepro temporis exponente scribi potest." 1 These are the propositions which HUYGHENS included, -without proof, at the end of his treatise. NEWTON 211 That is, " If a body P revolving about tke centre S, describes a curve line APQ, which a right line ZPR touches in any point P ; and from any other point Q of the curve, QR is drawn parallel to the distance SP, meeting the tangent in R ; and QT is drawn perpendicular to the distance SP — the centripetal force will be reciprocally as the solid SP2 • QT2 QR -, if the solid be taken of that magnitude which it ultimately acquires when the points P and Q coincide. " For QR is equal to the versed sine (sagitta) of double the arc ()P, whose middle is P — and double the triangle SQP, or SP X QT is pro portional to the time, in which that double arc is described ; and there fore may be used for the exponent of the time. " No purpose is served in indicating the generality and the remarkably direct character of this argument. The quantity QR is now called, in kinematics, the deviation. * dt2 This deviation has the value y — , where y is the acceleration. Now 2 since, here, the acceleration is central, like the force, and since it passes through the pole S, it is seen that QR is parallel to SP. Since the law of areas is applicable, the area of the triangle SPQ is proportional to dt. Since the force is itself proportional to the acceleration it is therefore, SP2-QT2 in the last analysis, inversely proportional to the expression . QR Fig. 79 212 THE FORMATION OF CLASSICAL MECHANICS Newton applies this general law to trajectories which are conic sections. We shall confine ourselves here to the elliptic trajectory and shall summarise the solution of Problem VI, proposition XI — " Revol- vatur corpus in ellipsi : requiritur lex vis centripetae tendentis ad umbi- licum ellipseos. " The original solution has, to some extent, the character of a rebus — we shall attempt to distinguish the essential steps and to express them in a way that will make this argument clearer. If DK is the diameter conjugate to CP, Newton first verifies that PE = a. Then drawing the line Qxv parallel to the tangent, cutting QR PE a SP and PC in x and v9 he verifies that -=j— = -5— = -^ because of the JT V Jrd Jr(^i similar triangles. In the same way, if QT is perpendicular to SP and PF is perpendicular to the tangent, Qx _ a Further, by Apollonius' Theorem _o_ CD PF~~ b " Hence Now in the limit when Q tends to P, — tends to unity. Therefore Qx We form the expression ,. - .. ^2 hm — pr— — = hm SP2-^= QR By the ecpiation of the ellipse referred to oblique conjugate axes CD and CP CD2 CP2 whence Qv 2 = CP2 - Cv2 _ (CP+Cv)Pv Cl)2 ~~ CP2 ~ CP2 NEWTON 213 Therefore limJg^^lim;^^ QR CP*-a-Pv aPC* * Whence the conclusion — 2J2 " Vis centripeta reciproce e$t ut — • SP2, id est reciproce in ratione a duplicata distantiae SP. " The law of force is inversely proportional to the distance. In short, this proof rests on the newtonian definition of force ; on the use of the kinematic idea of deviation ; and on a direct argument of infinitesimal geometry making use of the classical properties of conies. Except for the finite properties of conies, all its steps were unknown to Newton's predecessors, and were indispensible for the justification of Kepler* semi-empirical laws and for the fashioning of celestial mechanics into a chapter of dynamics. 7. THE UNIVERSAL ATTRACTION. The scope of this work does not allow us to deal with the numerous problems that are treated in the Principia. We shall only describe the path that was travelled by Newton's predecessors, and by Newton himself, and which ended in the law of universal attraction.1 Only an excessive schematisation can make the spontaneous blossom ing of a physical theory credible. The fall of an apple did not suffice to give Newton the idea of universal gravitation — rather, this was the product of a long development. As Early as the Xlllth Century Pierre de Maricourt, in a letter written in castris in 1269, analysed the polarities of a magnet in a very detailed way — the magnetic property tends to conserve the integrity of the magnet by binding its parts together. We have seen how the Schools of the XlVth Century, in the persons of Jean de Jandun, William of Ockham and Albert of Saxony, discussed the possibility of action at a distance.2 We have seen how Copernicus maintained that gravity was only a " natural desire " given to the parts of the Earth in order that their integrity might result.3 In De sympathia et antipathia rerum (1555), Frascator held that when two parts of the same whole were separated from each other, each of 1 For farther details the reader should consult DUHEM'S Tkeorie physique (Paris, ">), pp. 364 et se 2 See above, p 3 Ibid., p. 85. 1906), pp. 364 et seq. 2 See above, pp. 57-58. 214 THE FORMATION OF CLASSICAL MECHANICS them emitted a species which was propagated in the intermediate space. In De Magnete (London, 1600), Gilbert argued that the rectilinear motion of heavy bodies was the motion of the reunion of separated parts. He added that " this motion, which is only the inclination towards its source, does not only belong to the parts of the Earth, but also to the parts of the Sun, the Moon, and to those of the other celestial orbs. " Here Gilbert enters on the metaphysical plane. " We give the cause of this coming together and this motion which touches all nature. ... It is a substantial form which is special, particular, belong ing to primary and principal spheres ; it is a proper entity and an essence of their homogeneous and their uncorrupted parts, which we call a pri mary, radical and astral form ; it is not Aristotles' first form, but that special form by which the orb conserves and disposes what is its own. . . . It constitutes that true magnetic form that we call the primary energy. " 1 This animist philosophy, as Duhem has called it, was adopted by Francis Bacon. Kepler himself was stimulated by it, but substituted in it the idea on one single property belonging to any part of any star. We have already remarked 2 that Kepler regarded gravity as a " mutual affection between parent bodies that tends to unite them. " As far as the tides are concerned, Ptolemy had already produced an explanation by invoking a special influence of the Moon on the seas. In order to get rid of what seemed to them an occult quality, Averroes, Albertus Magnus and Roger Bacon attributed this action to the heat of the light from the Moon. Albert of Saxony championed an animist theory of the tides. Cardan, followed by Scaliger, believed only in an obedience of the waters to the Moon. Kepler himself wrote, " Observation proves that everything that contains humidity swells when the Moon waxes and shrinks when the Moon wanes. " 3 But later he corrected this opinion, and thereby anti cipated the Newtonian thesis. " The Moon acts not as a moist or damp star, but as a mass similar to the mass of the Earth. It attracts the waters of the sea, not because they are fluids but because they are gifted with terrestrial substance, to which they also owe their gravity. " 4 This attraction is reciprocal. " If the Moon and the Earth were in no way held by a sensual force or by some equivalent force, each in its orbit, the Earth would rise towards the Moon and the Moon would descend towards the Earth until these two stars joined together. If the 1 Translated into French by DUHEM. 2 See above, p. 118. 3 Opera omnia, Vol. I, p. 422. 4 Ibid., Vol. VII, p. 118. NEWTON 215 Earth ceases to attract the waters that cover it to itself, the waves of the sea would all rise and run towards the body of the Moon. " * Returning to a thesis which had already been put forward by Calcagnini, Galileo held that the ebb and the flow of the sea was explain ed by the following relative motion. The Earth turns from East to West at the same time that it is animated by a translational velocity v. At a the two motions add together — at 6, they tend to cancel out. Because of their inertia, the waters of the sea do not follow this motion exactly. The ebb and flow, thanks to the delay, is produced twice a day although, if the composition of the motions were perfect, they would have the period of the rotation of the Earth. Therefore Galileo interpreted the tidal phenomena as proof of the motion of the Earth, while the opponents of the copernican system held to a lunar attraction. The astrologers of the XVIth Century, following Grisogone, were inspired to separate the whole tide into a solar tide and a lunar tide. In 1528 Grisogone wrote, " The Sun and the Moon attract the swelling of the sea towards themselves, so that the maximum swelling is perpen dicularly beneath each of them. Therefore, for each of them, there are two maxima of swelling, one beneath the star and one on the opposite side, which is called the nadir of the star. " Ideas on the law of attraction itself were yet more vague and chan geable. To Roger Bacon, all actions at a distance were propagated in straight rays, like light. Kepler took up this analogy — now, it has been known since the time of Euclid that the intensity of the light emitted by a source varies in inverse ratio of the square of the distance from the source. In this optical analogy, the virtus movens emanating from the Sun and acting on the planets must follow the same law. But in dyna mics Kepler remained an Aristotelian — force was, to him, proportional to velocity. Therefore Kepler deduced the following result from the law of areas rv = constant. The virtus movens of the Sun on the planets is inversely proportional to the distance from the Sun. In order to reconcile this law with the optical analogy Kepler held that light spread out in all directions in space, while the virtus movens was only effective in the plane of the solar equator. Boulliau, writing Astronomia Philolaica in 1645, carried the optical 1 Opera omnia, Vol. Ill, p. 151. 216 THE FORMATION OF CLASSICAL MECHANICS analogy to its limit and supported the law of attraction inversely pro portional to the square of the distance. But it should be remarked that this attraction was normal to the radius vector, and not central as the Newtonian theory demanded. Descartes confined himself to replacing Kepler's virtus movens by a vortical ether. BoreUi has the merit of having invoked the example of the sling in order to explain why the planets did not fall on the Sun— he sets the instinct by which the planet carries itself towards the Sun against the tendency of aU bodies in rotation to move away from their centre— this vis repellens is inversely proportional to the radius of the orbit. In a paper caUed An Attempt to prove the annual Motion of the Earth (1674) Hooke, curator of the Royal Society, clearly formulated the prin ciple of universal gravitation. " AU celestial bodies without exception exert a power of attraction or heaviness which is directed towards their centre ; in virtue of which they not only retain their own parts and prevent them from escaping, as we see to be the case on the Earth, but also they attract all the celestial bodies that happen to be within the sphere of their activity. Whence, for example, not only do the Sun and the Moon act on the progress and motion of the Earth in the same way that the Earth acts on them, but also Mercury, Venus, Mars, Jupiter and Saturn have, because of their attractive power, a considerable influence on the motion of the Earth in the same way that the Earth has an influence on the motion of these bodies. " Hooke assumed that the attraction decreased with the distance and, in 1672, declared him self for the inverse square law. No doubt he was guided by the optical analogy. In order to justify this result it was necessary to know the laws of centrifugal force. Now we know that although Huyghens had written his treatise De vi centrifuga as early as 1659, only the statements of the thirteen propositions that conclude the Horologium oscillatorium were published during his lifetime. Halley appears to have applied Huyghens' theorems to Hooke's hypothesis. By assuming Kepler's third law (— = constant) he dis covered the law of the inverse square. This whole development, that we have only been able to summarise, shows that one cannot talk of the spontaneous generation of the theory of gravitation. For his part, Newton was in possession of the laws of uniform circular motion in 1666. By an analysis analogous to that which Halley had made, and starting from Kepler's third law, he formulated the law of NEWTON 217 an attraction inversely proportional to the square of the distance. But more careful than his predecessors, Newton sought experimental veri fication for this law. He tried to discover whether the attraction exert ed by the Earth on the Moon corresponded to this law, and whether this attraction could be identified with terrestrial heaviness. Since the radius of the Earth's orbit is of the order of 60 terrestrial radii, the force that maintains the Moon in its orbit is 3600 times weaker than the heaviness at the centre of the Earth. Now a body falling freely in the neighbourhood of the Earth falls a distance of 15 Paris feet l in the first second. The Moon would therefore fall a distance of — pouce in the first second. Knowing the period of the Moon's motion and the radius of its orbit, it is easy to calculate this fall of the Moon, With the data on the Earth's radius that were accepted in England, Newton obtained a fall of only — pouce. 2>& Faced with this divergence, he gave up his idea. It was only 16 years later (1682) that he learnt of the measurement of the terrestrial meridian that had been made by Picard. (This happened at a meeting of the Royal Society.) By assuming the value given by this determin ation, Newton obtained the expected value of — pouce. He was then 250 able to declare, " Lunam gravitare in Terrain et vi gravitatis retrahi semper a motu rectilineo et in orbe suo retineri " ; and, by an induction conforming to the very principles of his philosophy, to affirm the doctrine of universal gravitation. The theory of the attraction of spheres allowed him to concentrate at their centres the masses of stars that were supposed to be formed of homogeneous concentric layers, and thus to reduce them to material points whose mutual attractions could be studied. Newton evaluated the masses and densities of the Sun and the planets that were surrounded by satellites. He also calculated the heaviness at a point on their surface. He showed that the comets described very elongated elliptical trajectories and replaced these by parabolas whose elements he calculated. In this way he was able to connect the segments of trajectory of a comet that had appeared on each side of the Sun in 1680. Halley then showed that the appearances in 1531, 1607 and 1682 were those of this same comet. Newton also showed that the rotation of the Earth must entail its flattening at the two poles, and calculated the variation of gravity 1 In these discussions in the Priracipia, the distances are given in French units. 218 THE FORMATION OF CLASSICAL MECHANICS along a meridian. He related the theory of tides to the combined attraction of the Moon and the Sun and thus justified the anticipations of the astrologers of the XVIth Century. Finally, calculating the actions of the Moon and the Sun on the equatorial bulge, he arrived at a theory of the precession of the equinoxes. CHAPTER SEVEN LEIBNIZ AND LIVING FORCE 1. THE " vis MOTRIX " m THE SENSE OF LEIBNIZ. Leibniz protested against the cartesian mechanics in a memoir which appeared in 1686 in the Ada eruditorum at Leipzig, under the title A short demonstration of a famous error of Descartes and other learned men, concerning the claimed natural law according to which God always preserves the same quantity of motion ; a law which they use incorrectly , even in mechanics. Leibniz set out to show that the vis motrix (or, in the words of the XVIIIth Century, the force of bodies in motion), was distinct from the quantity of motion in Descartes' sense. Like Huyghens, Leibniz assumes that a body < falling freely from a given height will acquire the " force " necessary to rise again to the same height, if the resistance of the medium is neglected and no external inelastic obstacle is ~ encountered. On the other hand, like Des- r cartes, he assumes that the same " force " (in ^-^ the modern sense of work) is needed to lift a body A, whose weight is one pound, to a height -p. ^ DC of four ells as to lift a body J3, whose weight is four pounds, to a height of one ell. In falling freely from the height CD the body A acquires the same " force " as the body B acquires in falling from the height EF. For when it has arrived at D, the body A has acquired the force that it needs to climb again to C, and the body J5, when it has come to F, has acquired the force needed to climb to E. By hypothesis, these two forces are equal. Now the quantities of motion of A and B are far from being equal. Indeed, Galileo's laws show that the velocity acquired in the free fall CD is twice the velocity acquired in the free fall EF. The quantity 220 THE FORMATION OF CLASSICAL MECHANICS of motion of A is then proportional to 1x2, while that of B is pro portional to 4x1, and is therefore twice that of A. This contradicts the cartesian thesis in which the quantity of motion is used to evaluate the " force. " Leibniz recognised that in simple machines (the lever, the windlass, the pulley, the wedge and the screw) the same quantity of motion tended to be produced, in one part and the other, when equilibrium obtained. " Thus it happens by accident that the force can be reckoned as the quantity of motion. But there are other instances in which this coincidence no longer exists. " And Leibniz concludes, " It should be said, therefore, that the forces are in compound proportion to the bodies (of the same specific weight or density) and the generating heights of the velocities — that is, the heights from which the bodies are able to acquire their velo cities in falling, or more generally (since often no velocity has been produced at this point), the heights that will be generated. " x 2. LEIBNIZ AND THE LAWS OF IMPACT. Writing to the Abbe de Conti in 1687, Leibniz suggested that for the cartesian principle of the conservation of the quantity of motion should be substituted a natural law which he took as universal and inviolate. This was, " that there is always a perfect equality between the complete cause and the whole effect. " In this connection he went on to discuss Descartes' third rule on the impact of bodies.2 " Suppose that two bodies, B and C, each weighing one pound and travelling in the same direction, collide with each other. The velocity of B is 100 units and that of C, 1 unit. Their total quantity of motion will be 101. But if C, with its velocity, can rise to a height of one pouce, the velocity of B will enable it to rise to a height of 10,000 pouces. Thus the force of the two united bodies will be that of lifting one pound to 10,001 pouces. Now according to Descartes third rule, after the impact the bodies will go together in company with a common velocity of 50 and a half. . . . But then these 2 pounds are only able to lift themselves to a height of 2550 pouces and a quarter, which is equivalent to lifting one pound to 5100 pouces and a half. Thus almost half the force will be lost according to this rule, without there being any reason and without its having been used for anything. " 1 Translated into French by JOUGUET. 2 See above, p. 162. LEIBNIZ 221 In this discussion lies the germ of the controversy about living forces that was to divide the geometers at the beginning of the XVIIIth Century, and to which we shall return. We know now that Descartes third rule is correct and is applicable to perfectly soft bodies (soft, in order that they should travel together after the impact). The total quantity of motion is conserved (no difficulty of sign occurs here) and a part of the living force is transformed into heat. 3. LIVING AND DEAD FORCES. Leibniz showed himself to be even more systematic in his Specimen dynamicum (1695). We shall pass over the several quantities that he introduced and only concern ourselves with the distinction between living forces and dead forces. 44 Force is twin. The elementary force, which I call dead because motion does not yet exist in it, but only a solicitation to motion, is like that of a sphere in a rotating tube or a stone in a sling. 44 The other is the ordinary force associated with actual motion, and I call it living. 44 Examples of dead force are provided by centrifugal force, by gravity or centripetal force, and by the force with which a stretched spring starts to contract. 44 But in percussion that is produced by a body which has been falling for some time, or by an arc which has been unbending for some time, or by any other means, the force is living and born of an infinity of continued impressions of the dead force. " Leibniz reproached the Ancients " for having had exclusively an understanding of dead forces, and for only having studied the first conatus [in Huyghens sense] of bodies to each other, even though the latter had not acquired an impetus [in the sense of quantity of motion] by the action of the forces. In modern language, Leibniz's assertion that the living force is born of an infinity of impressions of the dead force may be expressed by This leads to the fundamental law m — = F and identifies the dead at force as the static force. CHAPTER EIGHT THE FRENCH - ITALIAN SCHOOL OF ZACCHI AND VARIGNON 1. ZACCHI AND SACCHERI. LAMY AND THE COMPOSITION OF FORCES. We shall devote this chapter to a brief analysis of some works which can only appear as miniatures in comparison with those of Galileo, Huyghens and Newton. But in leaving the peaks on which the work of the creators of dynamics lies, we shall have a better appre ciation of the extent to which those dominated their own century. In a Nova de machinis philosophia (Roma, 1649) Zacchi was concern ed with an attempt to isolate the principles implicit in Aristotle's statics. Under the term virtus he confused the concepts of force and work, and thus misunderstood Descartes' principle. Father Fabri (1606-1688) was a teacher at the Jesuit College at Lyons and a friend of Mersenne. His Tractatus physicus de motu locali (1646) was a work on dynamics which took over the ideas of Jordanus and Albert of Saxony. Among the moderns it only makes mention of Galileo's statics — and this, as we have seen, was impregnated with the ideas of Aristotle. Father Lamy attacked Descartes in his Treatise on Mechanics (1679) and contested Stevin's argument on the inclined plane. He claimed that no thing proved that the lower part of the chain of balls hung symmetrically. We now know that this criticism is not justified and that if the number of balls is infinite the necklace outlines a perfectly symme trical catenary underneath the plane. In order to solve the problem of the equilibrium of a body on an inclined plane, Lamy preferred to return to the arguments of Bernardino Baldi and da Vinci. To set against this is a letter addressed by Lamy to M. de Dieulamant, an engineer at Grenoble, which is concerned with the law of the com position of forces and deserves a little of our attention. " 1. When two forces draw the body Z along the lines AC and JBC, which are called the lines of direction of the forces, it is clear that the THE FRENCH-ITALIAN SCHOOL 223 body Z will not travel on the line AC or on the line JBC, but on another line between AC and BC, say X. " 2. If the path X were closed then Z, which is forced to travel by this path, would remain motionless, so that the forces would be in equilibrium. Fig. 82 " 3. Force is that which can move things. Motions are only mea sured by the distances which they travel. Suppose then that the force A is to the force B as 6 is to 2. Then if A9 in a first instant, draws the body Z as far as the point E on its own, in the same instant B would only draw it as far as F I CF = - CE ). We have seen that Z cannot \ 3 ^ / go along AC or along BC. Thus it is necessary that in the first instant it should come to D, where it corresponds to E and to F — that is to say, where it has travelled the value of CE and of FC. . . . " This line X is related to the lines of direction of the two forces A and B in such a way that at any point from which two perpendiculars on the two lines are drawn, their relation to each other will be the reciprocal of that of the forces, or the relation of DE to DF. " Lamy's demonstration is very similar to that on Newton. The simultaneity (1687) of the two demonstrations makes it seem however, that they were independent of each other. On the other hand, Lamy was accused of plagiarism from Varignon, who published his plan for a new mechanics at the same time. Lamy vigorously defended himself 224 THE FORMATION OF CLASSICAL MECHANICS against this charge. If, like Duhem,1 we put the emphasis on the words " in a first instant, " it is reasonable to believe that Lamy used an argument which would have been acceptable in modern mechanics. On the other hand, Varignon — who only cared for statics, a branch of the subject in which he showed great skill — did not progress beyond Aristotle's dynamics. We must also say a little about Neostatique (1703), a rather original work due to Father Saccheri. Saccheri regarded the vis matrix as proportional to the impetus, the term which he used to denote the absolute value of the velocity. As he was not concerned with the impetus of a body, starting from rest, in the first instant, this rule is equivalent to that of Aristotle. However, Saccheri arrived at an accord with Newton's dynamics. Thus he called the oriented velocity the impetus vivus, and used the term impetus subnascens for a quantity which, for a body of weight p, reduced to the projection of the acceleration — on the tangent. In identifying the m impetus subnascens as the incrementum of the impetus vivus, he was able to write down the Newtonian law of motion. This illustrates the extent to which the language and the ideas of the XVIIth Century were confused. Father Ceva had drawn Saccheri's attention to the law of Beaugrand and Fermat which we have mentioned in connection with the contro versy on geostatics.2 This is the law of an attraction which is proportional to the distance. Saccheri had the merit of showing that, according to this law, the heaviness passed through a centre of gravity that was fixed in the body. Also, that a body falling freely from rest and subject to this law, arrived at the common centre of heavy bodies in a time which did not depend on its distance from the centre. 2. THE STATICS OF VARIGNON (1654-1722). Varignon produced his Project for a New Mechanics in 1687, and the New Mechanics or Statics only appeared posthumously in 1725. At the beginning of the Project Varignon acknowledged the influence of Wallis and that of Descartes. The latter had declared that it was " a ridicul ous thing to wish to use the argument of the lever in the pulley " ; Varignon persuaded himself that it was equally useless to treat the 1 0. S., Vol. II, p. 259. 2 See above, p. 166. THE FRENCH-ITALIAN SCHOOL 225 inclined plane by starting from the lever. Of a more practical mind than his predecessors, he attached more weight to a study of the modes of equilibrium than to its necessity, and reduced everything to the prin ciple of compound motions. " It seems to me that the physical reason for the effects that are most admired in machines is exactly that of compound motion. " It is important to remark that Varignon interpreted the composition of forces and of motions in Aristotle's sense, for he remained consciously faithful to aristotelian dynamics. Indeed, in his New Mechanics he wrote — " Axiom VI. — The velocities of a single body or of bodies of equal mass are as all the motive forces which are there used, or which cause these velocities ; conversely, when the velocities are in this ratio they are those of a single body or of bodies of equal masses. " To Varignon, all force is analogous to the tension of a thread. In the diagrams which appear in his books, the hands holding the threads materialise the powers. He neglects all friction and even heaviness, which he identifies with a tension. " Requirement II. — That it may be permissible to neglect the heaviness of a body and to consider it as if it had none ; but to regard it as a power which may be applied to the weight ; when it will be con sidered as weight, notice will be given. ..." Varignon starts from a general principle which he expresses in the following way. " Whatever may be the number of forces or powers, directed as may be chosen, that act at once on the same body, either this body will not be displaced at all ; or it will travel along one path and along a line which will be the same as if, instead of being pushed in this way, com pressed or drawn by all these powers at once, the body was only follow ing the same line in the same direction by means of a single force or power equivalent or equal to the resultant of the meeting of all those forces. " Therefore everything reduces to the determination of this resultant. And it is here that Varignon affirms his allegiance to the Ancients. " It is what we are going to find by means of compound motions known to the ancients and the moderns — Aristotle treats them in the problems of mechanics ; Archimedes, Nicodemus, Dinostratus, Diocles, etc. . . . have used them for the description of the spiral, the conchoid, the cissoid, etc. . . . ; Descartes used them to explain the reflection and refraction of light ; in one word, all mathematicians use compound motions for the generation of an infinity of curved lines, and all correct physicists for determining the forces of impact or of oblique percussions, 226 THE FORMATION OF CLASSICAL MECHANICS etc. . . . Thus I claim nothing but the principle I indicated nearly forty years ago, and that I use once more for the explanation of machines. " Given this, it is easy to see how Varignon reduced the composition of forces to that of velocities. The superiority of Varignon' s work in statics is a didactic one. He treats all simple machines in detail by means of the composition of forces alone — this by ingenious procedures that are still commonly used. In Duhem's opinion, it does not seem that the geometers of the XVIIth Century, and even of the XVIIIth Century, had attached any importance to the distinction that can now be made between the method of Newton and of Lamy on the one hand, and of Varignon on the other, in the matter of the proof of the rule of the parallelogram. " The pro positions that aristotelian dynamics, over a period of two thousand years, had made customary in physics were also familar to all minds. They were still invoked naturally on all occasions when conscience did not too violently conflict with the truths of the new Dynamics. When Varignon, in 1687, produced his Project of a New Mechanics, he took as his starting point axioms which were said to have been borrowed from Physica auscultatio or De Caelo ; but at the same time Newton and Lamy showed that the same consequences could be obtained from an accurate dynamics. " * 3. VARIGNON AND TORRICELLI'S LAW OF FLOW. In the second Book of the Principia Newton had undertaken a proof of Torricelli's law of flow. He remarked that a column of liquid falling freely in a vacuum assumed the shape of a solid of revolution whose meridian was a curve of the fourth degree. Indeed, the velocity of each horizontal slice is proportional to the square root of the height from which it has fallen. On the other hand, this same velocity is inversely proportional to the section of the column, and consequently to the square of the radius. In a vessel having this shape and kept filled with water, it is clear that each particle of the fluid has its velocity of free fall and that, in consequence, Torricelli's law is justified. Newton then imagines that in a cylindrical vessel whose base is pierced with a hole, the fluid separates into two parts. One, the cata ract, takes the shape of free fall of which we have spoken. The other remains motionless. It is easy to see that this solution contradicts the principles of hydrostatics. Varignon had the merit of giving TorricellFs law a more natural 1 0. S., Vol. II, p. 260. THE FRENCH-ITALIAN SCHOOL 227 explanation. He assumed that the water remained sensibly immobile up to the immediate neighbourhood of the hole. At that point each particle instantaneously received, in the form of a finite velocity, the effect of the weight of the fluid that was above it. It is easy to see, taking account of the quantity of water flowing out, that the quantity of motion thus created in each particle is proportional to the square of the velocity. If the weight of the column of water above it is pro portional to the height A, Varignon can retrieve Torricelli's law, h = kv2. Lagrange l criticised this argument by observing that the pressure cannot suddenly produce a finite velocity. This was a difficulty that could not detain Varignon, to whom all force was generated by velocity. But one can, like Lagrange, assume that the weight of the column acts on the particle throughout the time that it is leaving the vessel. If it is then assumed that the fluid remains sensibly stagnant in the very interior of the vessel, Torricelli's law can be verified. 1 MGcanique analytique, Section VI, part I — Sur les principes de Vhydrostatique. PART THREE THE ORGANISATION AND DEVELOPMENT OF THE PRINCIPLES OF CLASSICAL MECHANICS IN THE XVIIIth CENTURY CHAPTER ONE JEAN BERNOULLI AND THE PRINCIPLE OF VIRTUAL WORK (1717) DANIEL BERNOULLI AND THE COMPOSITION OF FORCES (1726) 1. JEAN BERNOULLI AND THE PRINCIPLE OF VIRTUAL WORK. Classical mechanics was born in the XVIIth Century. The organ isation and development of the general principles had still to be ac complished — this was to be the work of the XVIIIth Century. The achievements of Galileo, Huyghens and Newton appear rather as disjointed parts than as the continuous development of a single discipline. Their successors, on the other hand, were to participate in a collective labour which, in the hands of Lagrange, was to end in an ordered science whose form approached perfection. In the preceding parts of this book we have treated each author in isolation from his contemporaries, and have attempted to follow the chronological order. In order to analyse the collective work of the XVIIIth Century, it will be more satisfactory if we devote each chapter to an attempt to collect together the work of different men that was relevant to one single topic. Although, at the end of the XVIIth Century, Varignon had tried to found statics on the one law of the composition of forces, we see Jean Bernoulli, in a letter to Varignon himself (January 26th, 1717), taking up the generalisation of what was really the principle of virtual work. We have seen that this principle had been used implicitly as early as the Xlllth Century, by the School of Jordanus, and that later it had been affirmed by Descartes and Wallis. Jean Bernoulli wrote, in the letter to Varignon, " Imagine several different forces which act according to different tendencies or in different 232 THE PRINCIPLES OF CLASSICAL MECHANICS directions in order to hold a point, a line, a surface or a body in equi librium. Also, imagine that a small motion is impressed on the whole system of these forces. Let this motion be parallel to itself in any direction, or let it be about any fixed point. It will be easy for you to understand that, by this motion, each of the forces will advance or recoil in its direction ; at least if one or several of the forces do not have their tendency perpendicular to that of the small motion, in which case that force or those forces will neither advance nor recoil. For these advances or recoils, which are what I call virtual velocities, are nothing else than that by which each line of tendency increases or decreases because of the small motion. And these increases or decreases are found if a perpendicular is drawn to the extremity of the line of tendency of any force. This perpendicular will cut ofi" a small part from the same line of tendency, in the neighbourhood of the small motion, which will be a measure of the virtual velocity of that force. f C Fig. 83 44 For example, let P be any point in a system which maintains itself in equilibrium. Let F be one of the forces, which would push or draw the point P in the direction FP or PF. Let Pp be a short straight line which the point P describes in a small motion, by which the tendency FP assumes the position fp. Either this will be exactly parallel to FP, if the small motion is, at every point, parallel to a straight line whose position is given ; or it will make an infinitely small angle with FP when this is produced, and if the small motion of the system is made around a fixed point. Therefore draw PC perpendicular to fp and you will have Cp for the virtual velocity of the force F, so that F X Cp is what I call the energy. 44 Observe that Cp is either positive or negative. The point P is pushed by the force F. It is positive if the angle FPp is obtuse and JEAN AND DANIEL BERNOULLI 233 negative if the angle FPp is acute. But on the contrary, if the point P is pulled, Cp will be negative when the angle FPp is obtuse and positive when it is acute. All this being understood, I form the follow ing general proposition. " In all equilibrium of any forces, in whatever way they may be applied and in whatever direction they may act — through intermedia ries or directly — the sum of the positive energies will be equal to the sum of the negative energies taken positively. " Jean Bernoulli's statement is much more general than those of his predecessors. Nevertheless, it must be remarked that the virtual displacements that are contemplated reduce to translations or rotations, to displacements in which the system behaves as a solid. Displacements of this kind are not necessarily compatible with the constraints of the system — they do not necessarily include the most general virtual displacement which is compatible with the constraints. Jean Bernoulli's principle does not seem to have accomplished a modification of Varignon's point of view. The latter was content to verify the principle in a large number of examples, which he treated with the methods to which he was accustomed. 2. DANIEL BERNOULLI AND THE COMPOSITION OF FORCES. In a memoir which appeared in 1726, called Examen principiorum mechanicae et demonstrationes geometricae de compositione et resolutione virium, Daniel Bernoulli set out to show that the law of the composition of forces was of necessary, and not of contingent, truth. We shall find that Euler and d'Alembert had similar preoccupations in other fields. The search for such a separation of purely rational truths from those which are subject to the uncertainties of, and correction by, experiment, was ever present in learned minds throughout the XVIIIth Century. The question, by its nature, is illusory. But the influence of Bernoulli's demonstration remained alive, and even Poisson was sub ject to it in 1833. Bernoulli regarded the hypothesis of the composition of motions on which Varignon had based his statics to be of a contingent kind. But a necessary truth can arise from two contingent hypotheses. In particular, the necessary law of the composition of forces depends, not only on the contingent hypothesis of the proportionality of the forces to the velocities that they produce but also, on the following hypothesis — A force which acts on a body that is already moved by 234 THE PRINCIPLES OF CLASSICAL MECHANICS another force impresses the same velocity on the body as if the latter were at rest. Basically, the development of Bernoulli's demonstration is the following x — Hypothesis I. — The composition of forces is associative. Hypothesis II. — The composition of two forces in the same direction reduces to algebraic addition. Hypothesis III. — The resultant of two equal forces is directed along their internal bisector — " a metaphysical axiom that must be regarded as a necessary truth. " With this basis, Bernoulli shows that if three forces are in equilibrium, so too are three forces which are the multiples of the first by the same number. He then establishes that the resultant of two equal forces at right angles is the diagonal of the square that has these two forces as sides. He continues with a consideration of two unequal rectangular forces and finds that the resultant is equal to the diagonal of the rect angle of which these two components form two sides. He also discusses the direction of the resultant. Bernoulli then treats pairs of components forming a rhombus whose angle is equal to I — I I — L Then, in order, components forming \2 / \2/ any rhombus, a rectangle, and a parallelogram. 1 For further details, c/. JOUGUET, L. M., Vol. II, p. 58. CHAPTER TWO THE CONTROVERSY ABOUT LIVING FORCES We know that* as early as 1686,1 in criticising the Cartesian notion of the conservation of quantities of motion, Leibniz had suggested that the " force " acquired by a body falling freely should be evaluated by the height to which this body could rise. Thus a body whose velocity is twice that of another is endowed with a force that is four times a great. The Abbe de Catelan protested that the body effected this ascent in twice the time. To produce a quadruple effect in twice the time is not to have a quadruple force, but only one which is twice as great. A child, in time, and bit by bit, will carry a sack of corn weighing 240 pounds. All force will be infinite if no regard is paid to time. After much hesitation, Jean Bernoulli came round to the opinion of Leibniz. In 1724 the Academie des Sciences, without using the words living force, chose the subject of the communication of motion for competition. Father Maziere, an adversary of the doctrine of living forces, was the successful competitor, in spite of a contribution from Jean Bernoulli that defended Leibniz. In this debate MacLaurin, Stirling and Clarke were opposed by the supporters of Leibniz — s'Gravesande, Wolf and Bulfinger. Bernoulli believed that the law v = Js\/Ti was related to that of gravity and that it was not an independent a priori law. Bodies would rise to infinity if no cause prevented them. The limitation is due to gravity, whose reiterated obstacles consumed a body's force of ascent. Bernoulli made use of other examples, of which the following is typical. If a perfectly elastic sphere A, moving with the velocity AC, collides obliquely with an identical sphere which it projects in the direction CD, the body C will be displaced on CD with the velocity CD = BC, while the body A will continue its journey with the velocity CE = CB. 1 See above, p. 219. 236 THE PRINCIPLES OF CLASSICAL MECHANICS Now the sum of the forces after the impact must be the same as the sum of the forces before the impact. This would be impossible if the force were proportional to the velocity, for CE + CD >• AC. On the other hand, this relation is verified if the force is proportional to the square of the velocity, for AC* = CD2 + A B C Fig. 84 In a Dissertation of the Estimation and the Measurement of the Motive Forces of Bodies (1728) de Mairan, like the Abbe de Catelan, opposed the evaluation of the force that the followers of Leibniz had suggested. His premises were simple. " As soon as I conceive that a body may be in motion, I conceive of a force that makes it move [to be understood as the vis motrix or the force of a body in motion, and not the corresponding dead force, which is zero for uniform motion]. A uniform motion can never indicate to us another measure of the force than the product of the simple velocity and the mass, " Here is the argument — " A massive body having two units of velocity is in such a state that it can mount to a height that is four times as great as that to which a body with only one unit of velocity would mount. " This proportion implies common measure. This common measure is the time ; at least I can take the time or the times to be equal. . . . " Now given this, in the effects of a body which has twice as much velocity, I only find an effect which is double and not quadruple — a distance travelled which is double, and a displacement of matter which is double, in equal times. From which I conclude, by the very principle of the proportionality of causes to their effects, that the Motive THE CONTROVERSY ABOUT LIVING FORCES 237 Force is not quadruple but only double, as the simple velocity and not the square of the velocity. " And de Mairan adds, " Strictly speaking, the concept of motion only includes uniformity. All motion should, on its own, be uniform, just as it should be effected in a straight line ; the acceleration and retardation are limitations which are foreign to its nature, as the curve that it is made to describe is to its proper direction. . . . " It is not the distances travelled by the body in retarded motion that give the evaluation and the measure of the motive force, but rather, the distances which are not travelled, and which should be travel led, in each instant by uniform motion. These distances which are not travelled are proportional to the simple velocities. And therefore the distances which correspond to a retarded or decreasing motive force, in so much as it is consumed in its action, are always proportional to this force and to the motion of the body, just as much in retarded motions as in uniform motions. " To explain this " kind of paradox, " de Maixan considers the example of two bodies, A and B, which ascend along AD and Bd. The body A has two units of velocity and B has only one. " If nothing opposes its motive force, in the first ,. ^ time B will travel the two toises Bd without losing T any part of this force or any part of the unit of velocity which gives rise to it. But because the contrary impulsions of the heaviness, which are continually applied to it succeed in consuming this force and its velocity, and in completely stopping it, the body will only travel one toise in its retarded motion. " In the same way, A would travel four toises in the first instant. The impulsions of the heaviness g' make it fall back through one toise, so that it only travels three. These impulsions have consumed one unit of force and one unit of velocity, as for B. But A remains with one unit and, at C, it finds itself in the initial case of B. It therefore has what it needs to travel the two toises CE. But the impulsions of the heaviness oppose it and it only travels CD, being pulled back through the one toise ED." Thus the distance which is not travelled by B in the first instant is fid. In the first instant the distance not travelled by A is CD, and in the second, is DJ?. This discussion is interesting — its metaphysical content is so apparent that we shall not emphasise it. Supporters and adversaries of the doctrine of living forces opposed each other with examples of impacting bodies. 0 238 THE PRINCIPLES OF CLASSICAL MECHANICS Thus Herman considers a perfectly elastic body M, of mass 1 and velocity 2, colliding with a motionless sphere N of mass 3. The body N will take, after the impact, the velocity 1 while the body M will be thrown back with the velocity 1. If M then meets a motion less body 0 of mass 1, it can communicate its velocity to the latter and remain at rest. Therefore the force of M, which has mass 1 and velocity 2, is equivalent to four times the force of a body of mass 1 and velocity 1, which verifies the law of living forces and contradicts that of quantities of motion. De Mairan observed that this coincidence was accidental and stemmed from the equality 2 + 2 = 2x2. For his part, he considered a body M of mass 1 and velocity 4 which he arranged to collide with a body N of mass 3 which was initially at rest. If M communicates a velocity 2 to JV, the force of N is as 6. The body M, which keeps the velocity 2, can transfer this to a body 0 of mass 1, initially at rest. The total force of M is therefore as 6 + 2 = 8, and not as 16 as the law of living forces would require. The Marchioness of Chatelet came round to the doctrine of living forces rather late in the day, and added an erratum to her book on the nature of fire (1740). While Koenig was a supporter of Leibniz, Maupertuis and Clairaut remained indifferent to this controversy. In the meantime, de Mairan tried to convince the Marchioness of Cha telet and, in 1741, Voltaire himself proclaimed his doubts about the measure of living forces. The error of the Cartesians, which was corrected in the course of the controversy by de Mairan, was that of reckoning the quantity of motion as m|t;|, without regard to the direction of the velocities. The reader will easily verify, in all the examples which have been cited — which are examples of elastic impact — that if the direction is introduced, that is, if quantities of motion mv are considered, then the quantities £mt; and £mt;2 a*6 both conserved. Therefore the con troversy of living forces was based on a mis-statement of the doctrine. It rested on a misunderstanding concerning the definition of quantity of motion which, as d'Alembert observed, divided the geometers for more than thirty years. CHAPTER THREE EULER AND THE MECHANICS OF A PARTICLE (1736) Euler (1707-1783) was concerned with all branches of dynamics, and we shall have occasion to return to his work in different connec tions. For the moment, we shall confine ourselves to the basic ideas of his treatment of the dynamics of a particle. This is found in Mechanic^ sive motus scientia analytics exposita which was published in 1736. The very title is a programme. Euler had read the great creators of mechanics, especially Huyghens and Newton, and he set out to fashion mechanics into a rational science by starting from definitions and logically ordered propositions. He tried to demonstrate the laws of mechanics in such a way that it would be clear that they were not only correct, but also necessary truths. To Euler, power or force is characterised by the modification of the motion of a particle that is produced by it. A power acts along a definite direction at each instant. This is what Euler expresses in the following definitions. " Potentia est vis corpus vel ex quiete in motum perducens, vel motum ejus alter ans. " " Directio potentiae est linea recta secundum quam ea corpus movere conatur. " In passing we remark that, in Eider's work, the term " corpus " denotes a particle. In the absence of force a particle either remains at rest, or is animated with a rectilinear and uniform motion. Euler expresses this principle with the help of the concept of " force of inertia. " " Vis inertiae est ilia in omnibus corporibus insita facultas vel in quiete permanendi vel motum uniformiter in directum continuendi. " Euler believes that " the comparison and the measurement of different powers should be the task of Statics. " Euler's dynamics 240 THE PRINCIPLES OF CLASSICAL MECHANICS is therefore primarily based on the notion of force, which he borrows directly from, statics in accordance with Galileo's procedure. Euler attempted to show that the composition or the equivalence of forces in statics could be extended to their dynamical effects. In fact, he was here concerned with a postulate. He also distinguished between absolute powers, such as gravity, that acted indifferently on a body at rest or in motion, and relative powers, whose effects depended on the velocity of the body. As an example of such a power, he cited the force exerted by a river on a body — this force disappears when the velocity of the body is the same as that of the river. In order to determine the effect of a relative power, an absolute power is associated with it, at least when the body has a known velocity. We return to the vis inertiae in the sense that Euler used it. For any body, this is proportional to the quantity of matter that the body contains. 64 The force of inertia is the force that exists in every body by means of which that body persists in its state of rest or of uniform motion in a straight line. It should therefore be reckoned by the force or power that is necessary to take the body out of its state. Now different bodies are taken out of their state to similar extents by powers which are proportional to the quantities of matter that they contain. There fore their forces of inertia are proportional to these powers, and conse quently, to the quantities of matter. " Euler assigns the same vis inertiae to one body, whether it is at rest or in motion. For in both cases the body is subject to the same action and the same absolute power. Here we see a systematisation of Newtonian ideas. Basically Euler introduces the mass — in the guise of a logical deduction — by means of the physical assertion of proportionality between the powers necessary to produce a given effect and the quantities of matter. As an example of Euler's analysis, we shall give his treatment of the following problem. " Proposition XIV. — Problem. — Being given the effect of an absolute power on a particle at rest, to find the effect of the same power on the same particle^ when the latter is moving in any way. " The absolute power which is given will make a body A, initially at rest, travel the path dz = AC in the time dt. If A has the velocity c, in the absence of any power it will travel the path AB — cdt in the time dt. But the given power, being absolute, acts on A in motion in the THE MECHANICS OF A PARTICLE 241 same way as it acts on A at rest. Therefore the effect of the power is compounded with that of the velocity, and the body A comes to D, where BD = AC. A Fig. 86 Under the effect of the given power, the velocity of A will become AD A simple geometrical argument shows that dc = — cos BAG. dt Strictly speaking it would be more natural to regard the effect of the power as being the increase of the velocity between the time t and the time t + dt ; that is, to consider the quantity 2dc instead of the difference between the initial velocity c and the mean velocity of A during the time dt.1 Euler then studies the effect of a power B on a body when the effect of a power A on the same body is given. He concludes — " If a body is affected by many powers, at first it may be thought of as divided into as many parts, on each of which one of the powers acts. Then, when the different parts have been drawn by their respec tive powers for an element of time, it is imagined that they suddenly unite. When this is accomplished, the position of their reunion will be that at which the whole body would have arrived in the same time by the simultaneous action of all the powers. The truth of this princi ple can be illustrated by remarking that the parts of a body can be held together by very strong springs which though they act in an undefined manner, can be supposed to relax completely in the interval 1 C/. JOUGUET, L. M., Vol. II, p. 43. 16 242 THE PRINCIPLES OF CLASSICAL MECHANICS of time, and to contract suddenly with an infinite force, afterwards, in such a way that the conjunction of the separated parts takes no time. " Thus Euler's law of dynamics takes the form — The increase, dc, of the velocity is proportional to pdt, where p is the power acting on the body during the time dt. This applies to a single body ; if several bodies are considered simultaneously, their masses must be introduced. Therefore this law emphasises the impulse of the force during an elementary time, or the impulse that gives rise to an increase of momentum. Euler declared that this law was not only true, but also a necessary truth, and that a law identifying mdc as p*dt or as p^dt would imply a contradiction. Clearly this is an illusion of the author. Eider's treatise then continues with a study of a large number of problems. First he treats a free particle, and concludes with a particle bound on a curve or a surface, either in a vacuum or a resisting medium. His work was the first to merit, for the order and the pre cision of its demonstrations, the name of a treatise of rational mechanics. CHAPTER FOUR JACQUES BERNOULLI AND THE CENTRE OF OSCILLATION (1703) D'ALEMBERT'S TREATISE ON DYNAMICS (1743) 1. JACQUES BERNOULLI AND THE CENTRE OF OSCILLATION. In 1703 Jacques Bernoulli returned to the famous problem of the search for a centre of oscillation, and gave a solution of it which contained the germ of d'Alembert's principle. Jacques Bernoulli's paper was called " (General demonstration of the centre of balancing and of oscillation deduced from the nature of the lever. " He considers a lever which is free to turn about a point A and whose different arms carry weights or powers which act perpendicularly to the arms. If the powers are divided into two groups that act on the lever in opposite senses, and if the sum of the products of the arms of the lever and the powers has the same absolute value for each group, then the lever will remain in equilibrium. This had been shown by Mariotte in the Treatise on the percussion of bodies. Given this, let A represent the axis of suspension, and let AC and AD join A to two arbitrary elements of a compound pendulum (for simplicity assumed to be plane). Then let AM be the simple pendulum isochronous with the compound pendulum. ^ Consider the motion of the elements C, D and M of the compound pendulum. Their velocities are proportional to AC^ AD and AM. At each instant the gravity adds an impact or an impulse which is represented by MN9 CO, DP, " short vertical and equal lines. " Take NK, OT and PV perpendicular to the arcs MK, CT, DV. Bernoulli considers the " motions " MZV, CO, DP as being decom posed into motions MK and KN ; CT and TO ; DV and VP. The motions JfiCJV, TO, VP " distribute themselves over the whole axis A 244 THE PRINCIPLES OF CLASSICAL MECHANICS and there lose themselves completely. " Because of the isochronism of the points C, D and M, the motions MK, CT and VD suffer " some change. " If, for example, M comes to K (without alteration), then C comes to R and D to S, and the arcs MK, CR and DS will be similar. The effort of gravity acting on the point C is not exhausted at JR and " the remainder, RT, must be used to push the body D along VS. " But D itself resists as much as it is pushed, and everything happens as if D travelled to S — as if there were a force " which tries to repel it from S to V. " Fig. 87 To sum up, the lever CAD is in equilibrium under the action of weights like C, " pulling or pushing from one side with forces or velo cities .RT," and weights like D, pulling or pushing in the opposite sense. Therefore Bernoulli writes S (C X CA X RT) = S (D X AD X VS) and, from this, deduces the solution of the problem of the centre of oscillation. 2. THE INTRODUCTORY ARGUMENT OF D'ALEMBERT'S TREATISE ON DYNAMICS. The first edition of d'Alembert's Traite de dynamique is dated 1743. Here we shall follow an edition of 1758, which was corrected and added to by the author. In an introductory discussion, d'Alembert explains his philosophy JACQUES BERNOULLI AND D'ALEMBERT 245 of mechanics. The Sciences are divided into two groups — those which are based on principles which are necessarily true and clear in themselves ; and those which are based on physical principles, experimental truths, or simply on hypotheses. Mechanics belongs to the first category of purely rational sciences, although it appears to us as less direct than Geometry and Algebra. It has failed to clarify the mystery of impene trability, the enigma of the nature of motion, and the metaphysical principle of the laws of impact. . . . The best method of discussing any part of mathematics " is to regard the particular subject of that science in the most abstract and direct way possible ; to suppose nothing, and to assume nothing about that subject, that the properties of the science itself does not suppose. " D'Alembert sets out " to throw back the boundaries of mechanics and to smooth out the approach to it ... and, in some way, to achieve one of these objects by means of the other. That is, not only to deduce the principles of mechanics from the clearest concepts, but also to apply them to new ends. " He strives " to make everything clear at once ; both the futility of most of the principles that have so far been used in mechanics, and the advantage that can be obtained from the combin ation of others, for the progress of that Science. In a word, to extend the principles and reduce them in number. " The nature of motion has been much discussed. " Nothing would seem more natural than to conceive of it as the successive application, of the moving body to the different parts of infinite space. " But the Cartesians, " a faction that, in truth, now barely exists, " refuse to distinguish space from matter. In order to counter their objections, d'Alembert makes a distinction between impenetrable space, provided by what are properly called bodies, and space pure and simple, penetr able or not, which can be used to measure distances and to observe the motion of bodies. " The nature of time is to run uniformly, and mechanics supposes this uniformity. " This is Newtonian. "A body cannot impart motion to itself, " There must be an extern al cause in order to move it from rest. But " if the existence of motion is once supposed, without any other particular hypothesis, the most simple law that a moving body can observe in its motion is the law of uniformity, and consequently, this is that which it must conform to. ... Therefore motion is inherently uniform. " D'Alembert defines the force of inertia as the property of bodies of remaining in their state of rest or motion. Among the means that can alter the motion of a body, apart from constraints, he only allows two — impact (or impulse) and gravity (or, more generally, attraction). 246 THE PRINCIPLES OF CLASSICAL MECHANICS In this connection it seems that d'AIembert criticises the very prin ciple of Euler's mechanics. " Why have we gone back to the principle, which the whole world now uses, that the accelerating or retarding force is proportional to the element of the velocity ? A principle supported on that single vague and obscure axiom that the effect is proportional to its cause. " We shall in no way examine whether this principle is a necessary truth or not. We only say that the evidence that has so far been pro duced on this matter is irrelevant. Neither shall we accept it, as some geometers have done, as being of purely contingent truth, which would destroy the exactness of mechanics and reduce it to being no more than an experimental science. We shall be content to remark that, true or false, clear or obscure, it is useless to mechanics and that, consequently, it should be abolished. " This shows in what sense d'AIembert interpreted the task of making mechanics into a rational science, and the extent to which he valued his own principle. D'AIembert made appeal to a principle of the composition of motions, of which he intended to give simple evidence. When a body changes in direction, its motion is made up of the initial motion and an acquired motion. Conversely, the initial motion can be compounded of a motion which is assumed and a motion which is lost. D'AIembert established the laws of motion in the presence of any obstacle in the following way. The motion of the body before meeting the obstacle is decomposed into two motions — one which is unchanged, and another which is annihilated by the obstacle. If the obstacle is insurmountable, the laws of equilibrium are used. These laws are expressed by a relation of the kind m v mr v where v, v' are the velocities with which the masses m, m! tend to move. Only when there is perfect symmetry, or when m = m' v' = — v does the problem appear inherently clear and simple to d'AIembert, and he tries to reduce all other situations to this one. We have seen that this was an illusion which Archimedes had in his investigation of the equilibrium of the lever. And d'AIembert concludes, " The principle of equilibrium, together with the principles of the force of inertia and of compound motion, JACQUES BERNOULLI AND D'ALEMBERT 247 therefore leads us to the solution of all problems which concern the motion of a body in so far as it can be stopped by an impenetrable and immovable obstacle — that is, in general, by another body to which it must necessarily impart motion in order to keep at least a part of its own. From these principles together can easily be deduced the laws of the motion of bodies that collide in any manner whatever, or which affect each other by means of some body placed in between them and to which they are attached. " Lagrange said, and it is often repeated, that d'Alembert had reduced dynamics to statics by means of his principle. The last quotation shows clearly that d'Alembert himself did not accept such a simple interpreta tion. On the contrary, he stressed the fact that " the three principles of the force of inertia, of compound motion and of equilibrium are essentially different from each other. " D'Alembert's beliefs are thus clearly expressed in the first pages on his introduction. But he also made clear his view on the problems which were popular in his time. Above all, he intended to take account of motion without being concerned with motive causes ; he completely banished the forces inherent to bodies in motion, " as being obscure and metaphysical, and which are only able to cover with obscurity a subject that is clear in itself. " This is why d'Alembert refused " to start an examination of the celebrated question of living forces, which has divided the geometers for thirty years. " To him, this question was only a dispute about words, for the two opposing sides were entirely in agreement of the fun damental principles of equilibrium and of motion. Their solutions of the same problem coincided, " if they were sound. " D'Alembert also discussed the question of knowing whether the laws of mechanics are of necessary or contingent truth. This question had been formulated by the Academy of Berlin. In order that this question may have a meaning, it is necessary to dispense with " every sentient being capable of acting on matter, every will of intellectual origin. " It is said that d'Alembert rejected every finalist explanation involving the wisdom of the Creator — we shall return to this in connection with the principle of least action. To d'Alembert, the principles of mechanics are of necessary truth. " We believe that we have shown that a body left to itself must remain forever in its state of rest or of uniform motion ; that if it tends to move along the two sides of a parallelogram at once, the diagonal is the direction that it must take ; that is, that it must select from all the others. Finally, we have shown that all the laws of the communic ation of motion between bodies reduce to the laws of equilibrium, and 248 THE PRINCIPLES OF CLASSICAL MECHANICS that the laws of equilibrium themselves reduce to those of the equili brium of two equal bodies which are animated in different senses by equal virtual velocities. In the latter instance, the motions of the two bodies evidently cancel each other out ; and by a geometrical conse quence, there will also be equilibrium when the masses are inversely proportional to the velocities. It only remains to know whether the case of equilibrium is unique — that is, whether one of the bodies will necessarily force the other to move when the masses are no longer inversely proportional to the velocities. Now it is easy to believe that as soon as there is one possible and necessary case of equilibrium, it will not be possible for others to exist without the laws of impact — which necessarily reduce to those of equilibrium — becoming indeter minate. And this cannot be, since, if one body collides with another, the result must necessarily be unique, the inevitable consequence of the existence and the impenetrability of bodies. " 3. D'ALEMBERT AND THE CONCEPT OF ACCELERATING FORCE. Of all the causes that could influence a body, d'Alembert was of the opinion that only impulse (that is, impact) was perfectly determinate. All other causes are entirely unknown to us and can only be distinguished by the variation of motion which they produce. The " accelerating force " 9? is introduced by the relation cpdt = du, a relation between the time t and the velocity u — the only observable kinematic quantities. This relation is the definition of (p. Therefore, to d'Alembert, this force was a derived concept, though to Daniel Bernoulli and Euler it constituted a primary concept. To Daniel Bernoulli, the law cpdt = du was a contingent truth ; to Euler, a necessary truth. D'Alembert wrote, " for us, without wishing to discuss here whether this principle is a necessary or a contingent truth, we shall be content to take it as a definition, and to understand by the phrase 6 accelerating force ', merely the quantity to which the increase in velocity is propor tional. " i 4. D'ALEMBERT'S PRINCIPLE. D'Alembert's principle was made the subject of a letter to the Academic des Sciences as early as 1742. In this book, we shall follow the presentation of the principle which appears in the 1758 edition of the Traite de Dynamique (2nd Part, Chapt. I, p. 72). 1 Trait^ de Dynamique^ cor. VI, p. 25 (1758 edition). JACQUES BERNOULLI AND D'ALEMBERT 249 PRESENTATION OF THE PRINCIPLE " Bodies only act on each other in three ways that are known to us — either by immediate impulse as in ordinary impact ; or by means of some body interposed between them and to which they are attached ; or finally, by a reciprocal property of attraction, as they do in the Newtonian system of the Sun and the Planets. Since the effects of this last mode of action have been sufficiently investigated, I shall confine myself to a treatment of bodies which collide in any manner whatever, and of those which are acted upon be means of threads or rigid rods. I shall dwell on this subject even more readily because the greatest geometers have only so far (1742) solved a small number of problems of this kind, and because I hope, by means of the general method which I am going to present, to equip all those who are familar with the calcu lations and principles of mechanics so that they can solve the most difficult problems of this kind. DEFINITION " In what follows, I shall call motion of a body the velocity of this same body and shall take account of its direction. And by quantity of motion, I shall understand, as is customary, the product of the mass and the velocity. GENERAL PROBLEM " Let there be given a system of bodies arranged in any way with respect to each other ; and suppose that a particular motion is imparted to each of these bodies, which it cannot follow because of the action of the other bodies — to find the motion that each body must take. SOLUTION " Let A9 JB, C, etc be the bodies that constitute the system and suppose that the motions a, 6, c, etc. . , . are impressed on them ; let there be forces, arising from their mutual action, which change these into the motions a, I>, c, etc. ... It is clear that the motion a impressed on the body A can be compounded of the motion a which it acquires and another motion a. In the same way the motions fr, c, etc. . . . can be regarded as compounded of the motions 1> and /?, c and #, etc. . . . From this it follows that the motions of the bodies -4, B, C, etc. . . . would be the same, among themselves, if instead of their having been 250 THE PRINCIPLES OF CLASSICAL MECHANICS given the impulses a, 6, c, etc. . . . they had been simultaneously given the twin impulsions a and a, b and /?, c and #, etc. . . . Now, by sup position, the bodies A^ B, C, etc. . . . have assumed, by their own action, the motions a, b, c, etc. . . . Therefore the motions a, /?, «, etc. . . . must be such that they do not disturb the motions a, b, c, etc. ... in any way. That is to say, that if the bodies had only received the mo tions a, /?, X, etc. . . . these motions would have been cancelled out among themselves, and the system would have remained at rest. 44 From this results the following principle for finding the motion of several bodies which act upon each other. Decompose each of the motions a, &, c, etc. . . . which are impressed on the bodies into two others, a and a, b and /?, c and X, etc. . . . which are such that if the motions a, I), c, etc. . . . had been impressed on the bodies, they would have been retained unchanged ; and if the motions a, /?, X, etc. . . . alone had been impressed on the bodies, the system, would have remained at rest. It is clear that a, b, c, etc. . . . will be the motions that the bodies will take because of their mutual action. This is what it was necessary to find. " 5. D'ALEMBERT'S SOLUTION OF THE PROBLEM OF THE CENTRE OF OSCIL LATION. Although d'Alembert's principle is perfectly clear, its application is difficult, and the Traite de Dynamique remains a difficult book to read. As a concrete example of its application, we C shall give d'Alembert's solution of the celebrat ed problem of the centre of oscillation.1 46 Problem. — To find the velocity of a rod CR fixed at C, and loaded with as many weights as may be desired, under the supposition that these bodies, if the rod had not prevented them, would have described infinitely short lines AO, BQ, jRT, perpendicular to the rod, in equal times. "All the difficulty reduces to finding the line RS travelled by one of the bodies, J?, in the ._ . time that it would have travelled RT. For then _. 00 the velocities J3G, AM, of all the other bodies rig. oo i are known. 44 Now regard the impressed velocities, RT, BQ, AO as being composed of the velocities RS and ST ; EG and — GQ ; AM and — MO. By our principle, the lever CAR would have 1 Trait^ de Dynamique, p. 96. JACQUES BERNOULLI AND D'ALEMBERT 251 remained in equilibrium if the bodies jR, B, A had received the motions ST, — GQ, —MO alone. " Therefore A-MO.AC + B.QG-BC = R.ST-CR. " Denoting AO by a, BQ by 6, J?Tby c, (L4 by r, CB by r', CjR by Q and JRS by 2, we will have R (c-*) Q=Ar- — a + Br' — - \Q / \ Q " Consequently _ Aarq + Bbr'g + RCQ* *~~ Ar* + Br'2 + RQ* " " Corollary. — Let .F,/, <p be the motive forces of the bodies A, B, The accelerating force will be found to be Fr+fr' + w Ke + Br2 +RQ - . F f <P on giving a, 6, c, their values -?, ~, — * Therefore, if the element of arc joL O /x described by the radius CR is taken to be ds and the velocity of R to be u, then, in general, whatever the forces F, /, cp may be. It is easy, by this means, to solve the problem of centres of oscillation under any hypothesis. 6. THE PRIORITY OF HERMAN AND EULER IN THE MATTER OF D'ALEM- BERT'S PRINCIPLE. After recalling Jacques Bernoulli's solution of the problem of the centre of oscillation, d'Alembert remarks that Euler, in Volume III of the old Commentaries of the Academy of Petersbourg (1740), had used the principle according to which the powers JR-JRS, B-J3G, A -AM must be equivalent to the powers JR-JRT, B-BQ, A-AO. "But M. Euler has in no way demonstrated this principle and this, it seems to me, can only be done by means of ours. Moreover, the author has only applied this principle to the solution of a small number of problems concerning the oscillation of flexible or inflexible bodies, and the solution that he has given to one of these problems is not correct. [This was the problem of the oscillation of a solid body on a plane.] This shows to what extent 252 THE PRINCIPLES OF CLASSICAL MECHANICS our principle is preferable for solving not only problems of tbat kind, but in general, all questions of dynamics. " 1 Lagrange had the following comment to make on this matter. " If it is desired to avoid the decompositions of motions that d'Alem- bert's principle demands, it is only necessary to establish immediately the equilibrium between the forces and the motions they generate, but taken in the opposite directions. For if it is imagined that there is impressed on each body the motion that it must take, in the opposite sense, it is clear that the system will be reduced to rest. Consequently, it is necessary that these motions should destroy those that the body had received and which they would have followed without this inter action. Thus there must be equilibrium between all these motions or between the forces which can produce them. " This method of recalling to mind the laws of Dynamics is certainly less direct than that which follows from d'Alembert's principle, but it offers greater simplicity in applications. It reduces to that of Herman 2 and of Euler,3 who used it in the solution of many problems in Mechan ics, and which is found in many treatises on mechanics under the name of d? Alember? $ Principle. " However clear these priorities may be, they do not detract from the originality of d'Alembert's conceptions. His work stands out because of its philosophic breadth of view, because of its property of unifying and generalising, and its equal is not found among the work of his immediate predecessors. 7. D'ALEMBERT AND THE LAWS OF IMPACT. D'Alembert systematically applied his principle to the solution of all the problems which appear in his Traite, whether they concern bodies which are supported by threads or rods, bodies which oscillate on planes, bodies which interact by means of threads on which they can run freely, or different modes of impact. In the problems of impact d'Alembert, at first, only considers *4 hard bodies " (that is, bodies deprived of their elasticity). Thus, if a body of mass M and velocity U collides directly with a body of mass m and velocity w, d'Alembert writes the following relations between the velocities. u — v + u — v u= v+ u-v 1 TraitS de Dynamique^ p. 101. 2 Phoronomia, sive De viribus et motibus corporum solidorum et fluidorum, Amster dam, 1716. 3 The paper cited by d'ALEMBERT (see the beginning of this §). JACQUES BERNOULLI AND D'ALEMBERT 253 Here v and V are the velocities of the first and the second bodies after the impact. After the impact V = v and because of the principle, m(u — v) -j- M(U— V) = 0. Therefore Fand v = *"" + MU. V ' M + 771 D'Alembert next deduces the laws of the impact of elastic bodies from those of the impact of hard bodies by the following procedure. " If as many bodies as may be desired collide with each other so that when it is supposed that they are perfectly hard and without elasticity, they all remain at rest after the impact ; I say that if they are of perfect elasticity each one will rebound after the impact with the velocity it had before the impact. For the effect of the elasticity is to give back to each body the velocity which it has lost because of the action of the others. " l Thus d'Alembert separated the theory of impact from all appeal to the conservation of living forces. 8. D'ALEMBERT AND THE PRINCIPLE OF LIVING FORCES. D'Alembert prepared the way for Lagrange by setting out to show that the principle of the conservation of living forces was a consequence of the laws of dynamics for systems with restraints composed of threads and inflexible rods, just as the laws of impact were a consequence of this same principle. Without giving a general demonstration of this fact, d'Alembert gave " the principles sufficient for obtaining the de monstration in every particular case. " We shall confine ourselves here to a very simple case. " Imagine two bodies, A and jB, of an infinitely small extension, to be attached to an inflexible rod AB. And suppose that any directions and velocities are imparted to these bodies, and that these velocities are represented by the infinitely short lines AK and J5D. According to our principle, it is necessary to construct the parallelograms MC and NL9 in which LC = AB and B X BM = A X AN. The velocities and the directions of the bodies B and A will be BC and AL. Now J3C2 = ED* — 2CE- CD— CD* and AD = Z8? + 2PL - KL — KL*. Therefore B.B& + A.JI* = A. AK* + B . HEM- AQPL-JSL — KL*± — B(2CE- CD + CD2), which reduces to A • AK* + B • BD*— A - KL* — B*CD\ since CE = PL and A-KL = B-CD. "Therefore 1 Traite de Dynamique, p. 218. 2 Ibid., p 253. CHAPTER FIVE THE PRINCIPLE OF LEAST ACTION 1. RETURN TO FERMAT. On January 1st, 1662, Fermat wrote to C. de la Chambre concerning refraction. " M. Descartes has never demonstrated his principle. For apart from the fact that comparisons are not of much use as foundations for demonstrations, he uses his own in the wrong way and even supposes that the passage of light is easier in dense bodies than in rare ones, which is clearly false. " In his investigation of the refraction of light, Fermat starts " from the principle , so common and so well-established, that Nature always acts in the shortest ways. " He first shows that in a parti cular numerical example, the recti linear path is not the most rapid for the traversal of two media by light. If the medium AGB is supposed to be more dense than the medium ACB, " so that the passage through the rarer medium is twice as easy as that through the denser one, " the time taken by the light in going from C to G by the straight line COG can FiS- 89 be represented " by the sum of half CO and the whole of OG. " Taking CO = 10, H 0 = OD = 8 and OF = 1, Fermat shows that +OG-15 CF = FG = ^/85 CQ and that, consequently,—- -f FG is less than — , and therefore less than 15. 2 4 THE PRINCIPLE OF LEAST ACTION 255 Fermat adds, " I arrived at this point without much trouble, but it was necessary to carry the investigation further ; and because, in order that my conscience might be satisfied, it was not sufficient to have found a point such as F through which the natural motion was accomplished more quickly, more easily and in less time than by the straight line COG, it was also necessary to find the point which allowed the passage from one side to the other in less time than any other there might be. In this connection, it was necessary to use my method of maximis and minimis, which is rather successful for expediting this kind of problem. " Fermat did not doubt the truth of his principle, but he had been warned from all sides that experiments confirmed Descartes' law. Therefore it was dangerous to try to introduce a " proportion different from those which M. Descartes has given to refractions. '* Moreover, it was necessary to " overcome the length and the difficulty of the calcu lation, which at first presented four lines by their fourth roots and accordingly became entangled in assymmetries. ..." However, his " passionate desire " to succeed fortunately inspired him to find a method which shortened his work by a half, in reducing these four asymmetries to only two. Fermat's calculation is found in his paper Synthesis ad refractiones, probably written in February, 1662. Fig. 90 Let there be a circle of diameter ANB, an incident ray MZV and a refracted ray NH. Let MRH be another trajectory passing through any point of AB, chosen, for example, on the right of IV. 256 THE PRINCIPLES OF CLASSICAL MECHANICS Fermat introduces the ratios velocity on MN _ MN __ velocity on MR _ MR * ' velocity on NH ~~ ~IN "" velocity on RH ~~ PJR > whence time on MN _ MN _IN_ IN time on NH ~ ~N3 ' MN ~ NH and, similarly, time on MR PR time on RH " so that time on MIVH = IN + NH time on MRH ~~ PR + RH' The point jff, " at which Nature herself takes aim " corresponds to a projection on AB such that = v ' ivs "ivi It is necessary to show that PR + RH > IN + NH. Immediately DIV MR, ... ..„. - = - by (1) and (2). NS PR ^ ' V ; Putting MN RN . DN NO and DJV NO NS NV ,„ f As DJV < MN, therefore JVO < RN ( ' \ As NS < DN, therefore NV< NO. Now, by (3), MR* = MN2 + NRZ + 2DN-NR = MN2 + NR* + 2MN-NO. Therefore, by (4) (5) MR > MN + NO. Now, by (1), (2), (3), DN = MN= NO = JVO + MJV MR NS IN NV NV + IN THE PRINCIPLE OF LEAST ACTION 257 Therefore, by (5), RP> NI + NV. It remains to prove that RH> HV for then it is clear that PR + RH > NI + NH. Now RH2 = NH2 + NR* — 2SN-NR and by (3) DN NO DN JWV NO Therefore HN NR NS NV or SN-NR = HN-NV since, by (4), NR > NV it follows that RH>NH—NV=HV, which completes the proof. We return to the letter which we quoted at the beginning of this section. Fermat concludes, " The reward of my work has been most extraordinary, most unexpected, and the most fortunate that I have ever obtained. For after having gone through all the equations, multi plications, antitheses and other operations of my method, and finally having settled the problem. . . , I found that my principle gave exactly the same proportion of the refractions that M. Descartes has established. I was so surprised by a happening that was so little expected that I only recovered from my astonishment with difficulty. I repeated my alge braic operations several times and the result was always the same, though my demonstration supposes that the passage of light through dense bodies is more difficult than through rare ones — something I believe to be very true and necessary, and something which M. Descartes believes to be the contrary. " What must we conclude from this ? Is it not sufficient, Sir, that as friends of M. Descartes, I might allow him free possession of his theorem ? Is it not rather glorious to have learned the ways of Nature in one glance, and without the help of any demonstration ? I therefore cede to him the victory and the field of battle. ..." 258 THE PRINCIPLES OF CLASSICAL MECHANICS 2. CARTESIAN OBJECTIONS TO FERMAT'S PRINCIPLE. Although his demonstration was mathematically incontestible, Fermat was not successful in convincing the Cartesians, who opposed it with metaphysical objections — which, at that time, took place over pure and simple reason. These facts emerge from the correspondence between Fermat and Clerselier. Thus Clerselier, writing to Fermat on May 6th, 1662, declares that Fermat's principle is, in his eyes, " a principle which is moral and in no way physical ; which is not, and which cannot be, the cause of any effect of Nature. " To Clerselier, the straight line is the only determinate — " this is the only thing that Nature tends to in all her motions. " And he explains — " The shortness of the time ? Never. For when the radius MN has come to the point JV, according to this principle it must there be indifferent to going to all parts of the circumference BHA, since it takes as much time to travel to one as to the other. And since this reason of the shortness of time will not, then, be able to direct it towards one place rather than towards another, there will be good reason that it must follow the straight line. For in order that it might select the point H rather than any other, it is necessary to suppose that this ray MN, which Nature cannot send out without an indefinite tendency towards a straight line, remembers that it has started from the point M with the order to discover, at the meeting between the two media, the path that it must then travel in order to arrive at H in the shortest time. This is certainly imaginary, and in no way founded on physics. " Therefore what will make the direction of the ray MN (when it has come to IV) change at the meeting with the other medium, if not that which M. Descartes urges ? Which is that the same force that acts on and moves the ray MIV, finding a different natural arrangement for receiving its action in this medium than in the other, one which changes its own in this respect, makes the direction of the ray conform to the disposition that it has at the time. " And Clerselier concludes — " That path, which you reckon the shortest because it is the quickest, is only a path of error and bewilderment, which Nature in no way follows and cannot intend to follow. For, as Nature is determinate in everything she does, she will only and always tend to conduct her works in a straight line. " As for the velocity of light in dense and rare bodies, Clerselier believed that it would be " clearly more reasonable " to accept Fermat's thesis. THE PRINCIPLE OF LEAST ACTION 259 But, with a fine assurance, lie writes, " M. Descartes — in the 23rd page of his Dioptrique — proves and does not simply suppose, that light moves more easily through dense bodies than through rare ones. " A letter from Fermat to Clerselier, dated May 21st, 1662, contains the following bitter ironical reply. " I have often said to M. de la Chambre and yourself that I do not claim and that I have never claimed, to be in the private confidence of Nature. She has obscure and hidden ways that I have never had the initiative to penetrate ; I have merely offered her a small geometrical assistance in the matter of refraction, supposing that she has need of it. But since you, Sir, assure me that she can conduct her affairs without this, and that she is satisfied with the order that M. Descartes has prescribed for her, I willingly relinquish my pre tended conquest of physics and shall be content if you will leave me with a geometrical problem, quite pure and in abstracto, by means of which there can be found the path of a particle which travels through two different media and seeks to accomplish its motion as quickly as it can. " Thus the problem was taken back on to the mathematical plane, the only profitable one. In a letter written in 1664 to an unknown person, Fermat returns to " the intrigue of our dioptrics and our refractions. " If one is to judge from the text, the Cartesians had not confessed themselves beaten. 46 The Cartesian gentlemen turned my demonstration, which was com municated to them by M. de la Chambre, upside down. At first they were of the opinion that it must be rejected, and although I represented to them very sweetly that they might be content that the field of battle should remain with M. Descartes, since his opinion was justified and confirmed, albeit by reasons different from his own ; that the most famous conquerors did not regard themselves less fortunate when their victory was won with auxiliary troops than if it was won by their own. At first they had no wish to listen to raillery. They determined that my demonstration was faulty because it could not exist without des troying that of M. Descartes, which they always understood to have no equal. . . . Eventually they congratulated me, by means of a letter from M. Clerselier. . . . They acclaimed as a miracle the fact that the same truth had been found at the ends of two such completely opposed paths and announced that they would prefer to leave the matter un decided, saying that they did not know, in this connection, whether to value M. Descartes' demonstration more highly than my own, and that posterity would be the judge. " 260 THE PRINCIPLES OF CLASSICAL MECHANICS In a paper in the Ada of Leipzig for 1682, Leibniz rejected Fermat's principle. Light chooses the easiest path, which must not be confused with the shortest path or with that which takes the shortest time. Leibniz contemplated a path of least resistance or, more accurately, a path for which the product of the path and the " resistance " might be a minimum. Leibniz also supported Descartes' opinion on the relative velocity of light in rare and dense bodies with the aid of the following arguments. Although glass " resists " more than air, light proceeds more quickly in glass than in air because the greater resistance prevents the diffusion of the rays, which are confined in the passage after the manner of a river which flows in a narrow bed and thus acquires a greater velocity. 4. MAUPERTUIS' LAW OF REST. Before coming to Maupertuis' dynamics, we shall devote a little attention to a law of mini mum and maximum which was put forward by this author in the Memoires de VAcademie des Sciences for 1740, and in which the concept of potential makes its appearance. 46 Let there be a system of bodies which gravitate, or which are attracted towards centres by the forces that act on each one, as the 7ith power of their distances from the centre. In order that all these bodies should remain at rest, it is necessary that the sum of the products of each mass with the intensity of the force 1 and with the (n + l)th power of its distance from the centre of its force (which may be called the sum of the forces at rest) should be a maximum or a minimum. " By means of this law of rest Maupertuis rediscovered the essential theorems of elementary statics (the rule of the parallelogram, the equilibrium of an angular lever). 5. THE PRINCIPLE OF LEAST ACTION IN MAUPERTUIS' SENSE (1744). The debate between Fermat and the Cartesians, and Leibniz's objections to Fermat's principle, prepared the way for Maupertuis' intervention. The latter stated the principle of least action in a paper read to the Academic des Sciences on April 15th, 1744. The paper is entitled The agreement between the different laws of Nature that had, until now, seemed incompatible. 1 The force is here of the form kmrn. THE PRINCIPLE OF LEAST ACTION 261 Haupertuis starts by recalling the laws which light must obey — rectilinear propagation in a uniform medium, the law of reflection and the law of refraction. He seeks simple mechanical analogies. " The first of the laws is common to light and to all bodies. They move in a straight line unless some outside force deflects them. 44 The second is also the same as that followed by an elastic ball which is thrown at an immoveable surface. 44 But it is also very necessary that the third law should be explained as satisfactorily. When light passes from one medium into another, the phenomena are quite different from those which occur when a ball is reflected from a surface which does not yield to it in any way ; or those which occur when a ball, on meeting one that does yield to it, continues its progress, only changing the direction of its path. . . . Several mathematicians have extracted some fallacy which had escaped the notice of Descartes, and have made the error of his explanation clear. " Newton gave up the attempt to deduce the phenomena of refrac tion from those which occur when a body encounters an obstacle, or when it is forced along in media that resist differently, and fell back on his attraction. Once this force, which is distributed through all bodies in proportion to the quantity of matter, is assumed, the phenomena of refraction are explained in the most correct and rigorous way. . . . 44 M. Clairaut, who assumes that light has a tendency towards transparent bodies, and who considers this to be caused by some atmo sphere which could produce the same effects as the attraction, has deduced the phenomena of refraction. . . . 44 Fermat was the first to become aware of the error of Descartes' explanation. . . . He did not rely on atmospheres about the bodies, or on attraction, although it is known that the latter principle was neither unknown nor disagreeable to him.1 He sought the explanation of these phenomena in a principle that was quite different and purely metaphysical. 44 This principle was 4 that Nature, in the production of her effects, always acts in the most simple ways. ' Therefore Fermat believed that, 1 MAUPERTUIS is here referring to a passage from FERMAT'S work (var. oper. p. 114) and which he cited elsewhere with the intention of showing that FERMAT had anticipated NEWTON. This does not seem very convincing, for FERMAT'S attrac tion remained metaphysical in essence. Here is this passage. " The common opinion is that gravity is a quality which resides in the falling body itself. Others are of the opinion that the descent of bodies is due to the attraction of another body, like the Earth, which draws those that descend towards itself. There is a third possibility — that it is a mutual attraction between the bodies which is caused by the mutual attraction that bodies have for each other, as is apparent for iron and a magnet. " 262 THE PRINCIPLES OF CLASSICAL MECHANICS in all circumstances, light followed at once the shortest path in the shortest time.1 This led him to assume that light moved more easily and more quickly in the rarer media than in those in which there is a greater quantity of matter. " When Maupertuis wrote it was generally agreed that light moved more quickly in denser media, in the manner specified by the newtonian law of the proportionality of the indices of refraction to the velocities of propagation. "All the structure that Fermat has built up is therefore destroyed In the paper that M. de Mayran has given on the reflection and re fraction, there can be found the history of the dispute between Fermat and Descartes, and the difficulty and inability there has so far been to reconcile the law of refraction with the metaphysical principle. " Therefore, unlike Fermat, Maupertuis sought a minimum principle that might be compatible with the newtonian law, and not with the now generally accepted law which goes back to Huyghens. The strange thing is not that he succeeded in finding it. Rather it is that, in boldly extending — one is even tempted to say gratuitously extending — this minimum principle into the field of dynamics, he was led to a law which was truly sufficient, and which he successfully opposed to the thesis of Descartes on the conservation of momentum and of Leibniz on the con servation of kinetic energy. Up to this point, our author has only criticised the different inter pretations of the laws of refraction that had been put forward. We shall now look at his achievement. The relevant passage merits quota tion in its entirely — on the rational plane, it would be impossible to conceal its extreme weakness. We now enter the metaphysical plane in the most complete sense of the word. " In meditating deeply on this matter, I thought that, since light has already forsaken the shortest path when it goes from one medium to another — the path which is a straight line — it could just as well not follow that of the shortest time. Indeed, what preference can there be in this matter for time or distance ? Light cannot at once travel along the shortest path and along that of the shortest time — why should it go by one of these paths rather than by the other ? Further, why should it follow either of these two ? It chooses a path which has a very real advantage — the path which it takes is that by which the quantity of action is the least. 1 As far as it concerns the path, this is incorrect. What is a minimum, to FERMAT, is the sum In -f I'n', the sum of the products of each trajectory with the corresponding refractive index in SNELL'S sense. THE PRINCIPLE OF LEAST ACTION 263 " It must now be explained what I mean by the quantity of action. When a body is carried from one point to another a certain action is necessary. This action depends on the velocity that the body has and the distance that it travels, but it is neither the velocity nor the distance taken separately. The quantity of action is the greater as the velocity is the greater and the path which it travels is the longer. It is propor tional to the sum of the distances, each one multiplied by the velocity with which the body travels along it.1 "It is the quantity of action which is Nature's true storehouse , and which it economises as much as possible in the motion of light. " Maupertuis' demonstration follows. " Let there be two different media, separated by a surface which is represented by the line CD, such that the velocity in the upper medium is proportional to m and the velocity in the lower medium is propor tional to n. Let there be a ray of light, starting from the given point A, which must pass through the given point B. In order to find the point JR at which it must break through, I seek the point at which, if the ray breaks through, the quantity of action is least. I have m • AR + n> • RB, which must be a minimum. 66 Or, having drawn the perpendiculars AC, BD, to the common surface of the two media, I have m VAC* + CR* + n VBD* + DR* = ff D C Fig. 91 or, since AC and BP are constants nPR-dDR __ VAC* + CR2 VBP* + 1 A footnote adds the following detail — " As there is only one body, the mass is neglected. " 264 THE PRINCIPLES OF CLASSICAL MECHANICS " But, since CD is a constant, there obtains dCR = — dDR. " Therefore mCR nDR , CR RD . . n and AR BR AR BR m or, in words, the sine of the incidence, or the sine of the refraction, are in inverse proportion to the velocity which the light has in each medium. " All the phenomena of refraction now agree with the great principle that Nature in the production of her works, always acts in the most simple ways. " Maupertuis then shows without difficulty that " this basis, this quantity of action that nature economises in the motion of light through different media, she also saves in the reflection and the linear propa gation. In both these circumstances, the least action reduces to the shortest path and the shortest time. And it is this consequence that Fermat took as a principle. " Maupertuis concludes, " I know of the repugnance that several mathematicians have for final causes when applied to physics, and to a certain extent I am in accord with them. I believe that they are not introduced without risk. The error, which men like Fermat and those that followed him, have committed, only shows that, too often, their use is dangerous. It can be said, however, that it is not the principle which has betrayed them, but rather, the haste with which they have taken for the principle what is merely one of the consequences of it. It cannot be doubted that all things are regulated by a Supreme Being who, when he impressed on matter the forces which denote his power, destined it to effect the doings which indicate his wisdom. " 6. THE APPLICATION OF THE PRINCIPLE OF LEAST ACTION TO THE DIRECT IMPACT OF TWO BODIES. In a paper published by the Royal Academy of Berlin in 1747, and called On the laws of motion and of rest, Maupertuis applied the principle of least action to the direct impact of two bodies. He only considered the effect of the direct impact of two homogeneous spheres, and started from the hypothesis that " the magnitude of the impact of two given bodies depends uniquely on their respective velocity, " that is, on their relative velocity. He distinguished between — " Perfectly hard bodies. These are those whose parts are inseparable and inflexible, and whose shape is consequently unalterable. THE PRINCIPLE OF LEAST ACTION 265 " Perfectly elastic bodies. These are those whose parts, after being deformed, right themselves again, taking up their original situation and restoring to the body its original shape. " Modern language would call the first category completely devoid of elasticity or perfectly soft. But the important matter here is that of the experimental laws which Maupertuis stated. "After the impact, hard bodies travel together with a common velocity. . . . The respective velocity of elastic bodies after the impact is the same as that before. " Maupertuis did not treat the intermediate case, " that of soft or fluid bodies, which are merely aggregates of hard or elastic ones. " He started from the principle that " when any change takes place in Nature, the quantity of action necessary for this change is the smallest possible. " We shall quote (notation apart) Maupertuis' argument on the impact of hard bodies. " Let there be two hard bodies A and B, whose masses are m and /ra', which move in the same direction with velocities v and v0 ; but A more quickly than B, so that it overtakes B and collides with it. Let the common velocity of the two bodies after the impact = vx < v0 and > VQ. The change which occurs in the Universe consists in that the body A, which used to move with a velocity VQ and which, in a certain time, used to travel a distance = t?0, now moves with the velocity i?! and travels no more than a distance = v±. The body -B, which only used to move with a velocity v'0 and travelled a distance = v^ moves with the velocity vl and travels a distance = v±. " This change is therefore the same as would occur if, while the body A moved with the velocity v0 and travelled a distance = t;0, it were carried backwards on an immaterial plane, which was made to move with a velocity t;0 — v^ through a distance = t;0 — i7x ; and that while the body B moved with the velocity v$ and travelled a distance = v& it were carried forwards by an immaterial plane, which was made to move with a velocity v± — VQ through a distance v± — VQ. " Now whether the bodies A and B move with their appropriate velocities on the moving planes, or whether they are immobile there, the motion of the planes loaded with these bodies being the same, the quantities of action produced in Nature will be m(vQ — t;x)2 and m/(vi — vo)2' an<^ fr *s the sum °£ these which must be as small as possible. Therefore it must be that mv\ — 2mvQv1 + mv\ + Hz't;f — 2mfv1VQ + mrv'^ = Min. or — 2mv0dv1 -\- 2mv1dvl + 2mfvidvl — ^m'v^dv^ = 0 266 THE PRINCIPLES OF CLASSICAL MECHANICS whence the common velocity mvn + mfv,( 1 m + m' is obtained. " No purpose would be served by reproducing the argument relevant to two bodies moving towards each other. Here too the condition of least action reduces to the conservation of the total momentum. Next treating the impact of elastic bodies, Maupertuis used an argument which was completely analogous to that which we have reproduced. Apart from sign, the " respective " velocity is conserved after the impact, or The cpiantity of action involved has the value and it follows from the condition of least action that _ __ 1 ~~ m + m 1 m + m' On this occasion the living forces are conserved, " but this con servation only takes place for elastic bodies, not for hard ones. The general principle, which applies to the first and to the others, is that the quantity of action necessary to produce some change in Nature is the smallest that is possible. " At the end of his paper, Maupertuis dealt with the principle of the lever, and deduced it from the principle of least action. " Let c be the length of the lever, which I suppose to be immaterial, and at whose ends are placed two bodies whose masses are A and B. Let z be the distance of the body A from the point of support which is sought, and c — % be the distance of the body B. It is clear that, if the lever has some small motion, the bodies A and B will describe small arcs which are similar to each other and proportional to the distances of the bodies from the point which is sought. Therefore these arcs will be the distances travelled by the bodies, and at the same time will represent their velocities. The quantity of action will therefore be proportional to the product of each body by the square of its arc. Or (since the arcs are similar) to the product of each body by the square of its distance from the point about which the lever turns, that is, to THE PRINCIPLE OF LEAST ACTION 267 Az~ and B(c — s)2, and it is the sum of these which must be the smallest possible. Therefore Az*+ B(c-z)* = Min. or 2Azdz + 2Bzdz — ZBcdz = 0 from which it is deduced that Be Z~~ A + B which is the fundamental proposition of statics. " 7. THE PRINCIPLE OF LEAST ACTION IN MAUPERTUIS' WORK. Maupertuis, who had been a musketeer, had a great liking for geo metry. He was a surveyor and, in an amateur way, a geographer, astronomer, biologist, moralist and linguist. . . . And to crown and grace it all, Maupertuis was a metaphysician. Although he had a systematic mind, because of a trait rather common to men of his pro vince he was not free from fantasy. From this fantasy, or perhaps from his temperament, sprang naivety. We shall therefore turn over the pages of Maupertuis' work, seeking an explanation of the principle of least action.1 Here we shall only dwell on the Essai de Cosmologie. In this docu ment Maupertuis contrasted the rationalist school, " wishing to submit Nature to a purely material regime and to ban final causes entirely, " with the school which, on the contrary, " makes continual use of these causes and discovers the intentions of the Creator in every part of Nature. . . . According to the first, the Universe could dispense with God ; according to the second, the tiniest parts of the Universe are as much demonstrations " of the existence of God. He declared, " I have been attacked by both these factions of philosophy. . . . Reason defends me from the first, an enlightened century has not allowed the other to oppress me. " Thus Maupertuis flattered himself with having found a happy mean between these two extreme attitudes. " Those who make immoderate use (of final causes) have wished to persuade me that I seek to deny the evidence of the existence of God — which the Universe everywhere presents to the eyes of all men — in order to substitute for it one which has only been given to a few. " 1 We have referred both to the Dresden edition (Walter, 1752) and the Lyons edition (Bruyset, 1756). 268 THE PRINCIPLES OF CLASSICAL MECHANICS Among the evidence of the existence of God, Maupertuis intended to dispense with all that was provided by metaphysics. He also took no account of that which sprang from the structure of animals and plants, such as the proof— to cite only one — offered by the folds in the skin of a rhinoceros, who would not be able to move without them. " Philosophers who have assigned the cause of motion to God have been reduced to this because they did not know where else to place it. Not being able to conceive that matter had any ability to produce, distribute and destroy motion, they have resorted to an Immaterial Being. But when it is known that all the laws of motion are based on the principle of better, it cannot be doubted that these have their found ation in an omnipotent and omniscient Being, whether he gave bodies the power to interact with each other, or whether he used some other means which is still less understood by us. " He was not concerned, like Fermat, with assuming that Nature acts in the most simple ways. He was not concerned, as Descartes was, with assuming that the same quantity of motion was always conserved in Nature — " He deduced his laws of motion from this ; observation belied them, for the principle was not true. " Finally he was not concerned, like Leibniz, in assuming that the living force was always conserved. Huyghens and Wren had discovered the laws of the impact of elastic bodies simultaneously, but Huyghens had not taken these laws onto the plane of a universal principle. The conservation of living forces does not apply to the impact of hard bodies and, on this occasion, Maupertuis accused the followers of Leibniz " of preferring to say that there are no hard bodies in Nature " than to give up their principle. " This has been reduced to the strangest paradox that love of a system could produce — for the primitive bodies, the bodies that are the elements of all others, what can they be but hard bodies ? " Therefore Maupertuis denied all general principles that were not final. " In vain did Descartes imagine a world which could arise from the hand of the Creator. (Strictly speaking, Descartes' system supposes the initial intervention of the Creator, and the continuance of his " customary assistance. ") In vain did Leibniz, on another principle, devise the same plan. " And he concludes, " After so many great men have worked on this matter, I hardly dare say that I have discovered the principle on which all the laws of motion are founded ; a principle which applies equally to hard bodies and elastic bodies; from which the motions of all corpo real substances follow. . . . Our principle, more in conformity with the ideas of things that we should have, leaves the world in its natural need of the power of the Creator, and is a necessary result of the wisest THE PRINCIPLE OF LEAST ACTION 269 doing of that same power. . . . What satisfaction for the human mind, in contemplating these laws — so beautiful and so simple — that they may be the only ones that the Creator and the Director of things has established in matter in order to accomplish all the phenomena of the visible world. " 8. D'ALEMBERT'S CONDEMNATION OF FINAL CAUSES. D'Alembert himself was not directly involved in the polemic on the principle of least action that we shall describe in the next section. But he did completely condemn the intervention of final causes in the prin ciples of mechanics. Indeed, he wrote,1 " The laws of equilibrium and of motion are necessary truths. A metaphysician would perhaps be satisfied to prove this by saying that it was the wisdom of the Creator and the simplicity of his intentions never to establish other laws of equilibrium and of motion than those which follow from the very existence of bodies and their mutual impenetrability. But we have considered it our duty to abstain from this kind of argument, because it has seemed to us that it is based on too vague a principle. The nature of the Supreme Being is too well concealed for us to be able to know directly what is, or is not, in conformity with his wisdom.2 We can only discover the effect of his wisdom by the observation of the laws of nature, since mathema tical reasoning has made the simplicity of these laws evident to us, and experiment has shown us their application and their scope. " It seems to me that this consideration can be used to judge the value of the demonstrations of the laws of motion which have been given by several philosophers, in accordance with the principle of final causes ; that is, according to the intentions that the Author of nature might have formulated in establishing these laws. Such demonstra tions cannot have as much force as those which are preceded and supported by direct demonstrations, and which are deduced from prin ciples that are more within our grasp. Otherwise, it often happens that they lead us into error. It is because he followed this method, and because he believed that it was the Creator's wisdom to conserve the same quantity of motion in the Universe always, that Descartes has been misled about the laws of impact.3 Those who imitate him 1 Traite de Dynamique, Discours preliminaire, 1758 edition, p. 29. 2 Clearly an allusion to MAUPERTUIS. 3 The reader knows that DESCARTES' error is not, in fact, that of having asserted the conservation of momentum, but of having considered m\v\ instead of mv. 270 THE PRINCIPLES OF CLASSICAL MECHANICS run the risk of being similarly deceived ; or of giving as a principle, something that is only true in certain circumstances ; or finally, of regarding something which is only a mathematical consequence of cer tain formulae as a fundamental law of nature. " 9. THE POLEMIC ON THE PRINCIPLE OF LEAST ACTION. In the Acta of Leipzig for 1751 Koenig, Professor at The Hague, reproduced part of a letter which he aUeged had been written by Leibniz to Herman in 1707, and which contained the following passage. "Force is therefore as the product of the mass and the square of the velocity, and the time plays no part, as the demonstration which you use shows clearly. But action is in no way what you think. There the consideration of the time enters as the product of the mass by the distance and the velocity, or of the time by the living force. I have pointed out that in the variations of motions, it usually becomes a minimum or a maximum. From this can be deduced several important propositions. It can be used to determine the curves described by bodies that are attached to one or several centres. I wished to treat these things in the second part of my Dynamics but I suppressed them, because the hostile reception with which prejudice, from the first, accorded them, disgusted me. " Maupertuis, for his part, represented the affair in the following way.1 " Koenig, Professor at The Hague, took it into his head to insert in the proceedings of Leipzig a dissertation in which he had two ends in view — rather contradictory ones for such a zealous partisan of M. de Leibniz, but which he found it possible to unite. . . . He attacked my principle as strongly as possible. And, for those that he was unable to persuade of its falsehood, he quoted a fragment of a letter from Leibniz from which it could be inferred that the principle belonged to that one. " Summoned by Maupertuis to produce the letter, Koenig referred him to " a man whose head has been cut off" (Henzi, of Berne). No trace of this letter was found in spite of all the searches ordered by the King at the request of the Academie. The matter became a very acri monious one. " It was no longer a matter of reasons. M. Koenig and his supporters only replied with abuse. Finally they resorted to libel. . . ." At the time, Maupertuis presided over the Academy of Berlin on the appointment of Frederic II. Koenig returned his diploma to the Academy and published an Appeal to the public from the judgement 1 (Euvres completes, 1756 edition, Letter XI. — Sur ce qui s'est passe d /'occasion du principe de moindre action. THE PRINCIPLE OF LEAST ACTION 271 that the Academy of Berlin had pronounced in this matter. In 1753 he emphasised this with a Defence of the Appeal to the public which he addressed to Maupertuis and which he claimed not only the priority of Leibniz, but also that of Malebranche, Wolf, s'Gravesande and Engel- hardt. Voltaire took part in the controversy. Maupertuis wrote,1 " The strangest thing was to see appear as an auxiliary in this dispute a man who had no claim to take part. Not satisfied with deciding at random on this matter — which demanded much knowledge which he lacked — he took this opportunity to hurl the grossest insults at me, and was soon to cap them with his Diatribe.2 I allowed this torrent of gall and filth to run on, when I saw myself defended by the pen and the sceptre. Although the most eloquent pen of all had uttered these libels, justice made his work burn on the gibbets and in the public places of Berlin. " " My only fault, " declared Maupertuis, " was that of having disco vered a principle that created something of a sensation. " Euler, director of the Academy of Berlin, presented the following report. " This great geometer has not only established the principle more firmly than I had done but his method, more ubiquitous and penetrating than mine, has discovered consequences that I had not obtained. After so many vested interests in the principle itself, he has shown, with the same evidence, that I was the only one to whom the discovery could be attributed. " 1 Ibid. 2 La Diatribe du Dr Akdkia^ medecin du Pope, is too well known to need emphasis here. We confine ourselves to the extraction of what is directly relevant to our subject. At the beginning VOLTAIRE writes, " We ask forgiveness of God for baving pretended tbat tbere is only proof of bis existence in A -f- B divided by Z, etc. ..." This is both a reference to the demonstration of the equilibrium of the lever by means of the principle of least action and to MAUPERTUIS' rejection, in his Essai de Cosmologie, of metaphysical proofs of the existence of God. Then, in the guise of SL Decision of the professors of the College of Wisdom^ VOLTAIRE makes, in spite of the malicious terms in which it is couched, an accurate criticism. " The assertion that the product of the distance and the velocity is always a minimum seems to us to be false, for this product is sometimes a maximum, as Leibniz believed and as he has shown. It seems that the young author has only taken half of Leibniz's idea ; and, in this, we vindicate him of ever having had an idea of Leibniz in its entirety." And finally, concerning the part played by EULER, which MAUPERTUIS had not thought of concealing, the same Decision declares, " We say that the Copernicus's, the Kepler's, the Leibniz's . . . are something, and that we have studied under the Bernoulli's, and shall study again ; and that, finally, Professor Euler, who was very anxious to serve us as a lieutenant, is a very great geometer who has supported our principle with formulae which we have been quite unable to understand, but which those who do understand have assured us they are full of genius, like the published works of the professor referred to, our lieutenant. ..." We must also add that MAUPERTUIS is caricatured in a consistently malicious way in Microm6ga$, Candide, and in ISHomme aux quarante ecus. 272 THE PRINCIPLES OF CLASSICAL MECHANICS 10. EULER'S JUDGEMENT ON THE CONTROVERSY ON LEAST ACTION. Traces of Eider's opinion about the controversy on least action can be found in a Dissertation on the principle of least action, with an exam ination of the objections to this principle made by Professor Koenig. This was printed at Berlin, in Latin and French, in 1753. Euler discloses a great respect for Maupertuis, our " illustrious President. " He pays homage to Maupertuis law of rest in the following terms. This principle indicates " the marvellous accord of the equili brium of bodies, whether rigid, flexible, elastic or fluid. From each attraction can be deduced the Efficacy of each force, and there is equili brium when the sum of all the efficacies is least. " Euler remarks, " Professor Koenig places us under the twin obli gation of proving that the principle of least action is true, and that it does not belong to Leibniz. " To Koenig, all instances of equilibrium can be deduced successfully from the principle of living forces. The " Koenigian principle " consists of " the annihilation of the living force if there were no equilibrium. " It can be seen, " more clearly than the day, " that where the applied forces produce no living force, there is equilibrium. In short, in stating the principle of the nullity of the living force Professor Koenig is " con cealing that which he found first, 6 that in the state of equilibrium there is neither motion nor living force '. " In this form, the principle of Koenig may appear a truism but, to be accurate, his method proceeds in the following way. First, the system is displaced from its equili brium position and the living force calculated. Then this is cancelled out and the conditions of equilibrium deduced. This method searches the difficulty, for the calculation of motion is, in general, more difficult than that of equilibrium. And Euler concludes, " Koenig's principle usually leads to great circumlocutions and is, often, incapable of appli cation. " To Koenig, action does not differ from living force. He considers himself able to assert that " It is clearly seen that all equilibrium arises from the nullity of the living force or from the nullity of the action, taken correctly, and in no way from their Min. of Max. " Euler forthrightly condemned this thesis and, in passing, made the following observation. " Professor Koenig seems too attached to metaphysical speculations to be able successfully to withdraw his mind from those subtle abstrac tions and to apply it to the ordinary and material ideas such as those which are the subject of mechanics. " THE PRINCIPLE OF LEAST ACTION 273 In the next section we shall study Euler's personal contribution to the extremum principle in dynamics. In the document which concerns us here, he only made the following allusion to this matter. 44 1 am not in any way concerned, here, with the observation which I have made in the motion of the celestial bodies and, general, of those attracted to fixed centres of force, that if the mass of the body is multi plied by the distance travelled, at each instant, and by the velocity, then the sum of all these products is always the least. " To Euler, the question is one of an a posteriori verification and not of an a priori deduction. Further, Euler acknowledged Maupertuis' priority in the principle of least action. " Since this remark was only made after M. Maupertuis had presented his principle, it should not imply any prejudice against his originality. " 11. EULER AND THE LAW OF THE EXTREMUM OF / mvds. As early as 1744 Euler published a work called Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Bousquet, Lau sanne). Here we are concerned with his Appendix II — De motu projectorum in medio non resistente per methodum maximorum ac mini- moTum determinando. Euler starts from the following principle. " Since all the effects of Nature obey some law of maximum or minimum, it cannot be denied that the curves described by projectiles tinder the influence of some forces will enjoy the same property of maximum or minimum. It seems less easy to define, a priori, using metaphysical principles, what this property is. But since, with the necessary application, it is possible to determine these curves by the direct method, it may be decided which is a maximum or a minimum. " x Euler emphasised, in the Dissertation which we have analysed in § 10, that the matter was, to him, one of the a posteriori verification of the existence of an extremum in particular examples of the dynamics of a particle. The quantity which Euler considered was, at first, Mds-\/~v. Here M is the mass of the particle, ds the element of distance travelled and v the height of fall. Since the velocity is -\/~v.> dt = ds : *\/~v9 and J ds <\/ v = J vdt. The first integral refers to momenta and the second to living forces. This duality enabled Euler to emphasise that he did not offend the feelings of any party to the controversy on living forces. 1 Translated into French by JOUGUET. 18 274 THE PRINCIPLES OF CLASSICAL MECHANICS Euler verified that the integral J ds <\/~v = J vdt is an extremum in the parabolic motion of a particle subject to a central force. He then generalised this result to a particle attracted by any number of fixed centres. Mach remarks in this connection, u Euler, a truly great man, lent his reputation to the principle of least action and the glory of his inven tion to Maupertuis ; but he made a new thing of the principle, practic able and useful. " One should observe, however, that Euler did not condemn the doctrine of final causes as d'Alembert had done. On the contrary, the true significance of an extremum principle should be, in his opinion, sought in a sound metaphysics. Indeed, he concludes in the following terms. „ Since bodies, because of their inertia, resist all changes of state, they will obey forces which act on them as little as possible if they are free. Therefore, in the motion generated the effect produced by the forces will be less than if the bodies were moved in some other way. The strength of this argument may not be sufficiently clear. If, however, it is in accord with the truth I have no doubt that a sounder metaphysics will enable it to be demonstrated clearly. I leave this task to others, who make a profession of metaphysics (quod negotium aliis, qui metaphysicam pro- faentur, relinquo) . " 12. FINAL REMARK. To recapitulate, Fermat, in geometrical optics, stated the first mini mum principle that was not trivial. He was not able to convince the Cartesians although he eventually accepted a reduction of his principle to the rank of a " small geometrical assistance " offered to Nature without any pretension to dictate her doings. No one accepted Format's conclusion, however plausible it might have been, on the relative velocity of light in dense and rare media. Maupertuis cannot be reproached for having shared the errors of his time, reinforced as they were by the double authority of Descartes and Newton. By means of a very simple differential argument, Maupertuis succeed ed in making both the newtonian law of propagation and that of refrac tion amenable to an extremum law. Was the development of his thought as was said at the time, of an exclusively metaphysical kind ? Yes, if the explanation of his motives with which he prefaced his analysis is considered on its own. I am reluctant to suggest a more natural, but much more worldly, explana tion—that Maupertuis had, in his presentation, reversed the order of THE PRINCIPLE OF LEAST ACTION 275 the arguments ; that he first discovered the differential argument which we have reproduced and then presented it, a posteriori, as the conse quence of an economic principle which indicated both the power and the wisdom of the creator. If this had been the whole of Maupertuis' contribution, his name would have fallen into oblivion, at least as far as the invention of prin ciple is concerned. For in optics only Fermat's principle, which Mau pertuis had set out to demolish, has survived. Maupertuis' extension of the principle of least action to dynamics appears rather gratuitous, for it rests on a fragile analogy — yet it is this principle which has survived and assured the fame of its author. Certainly, as early as 1744, Eider gave the exact mathematical justifi cation of the principle in the special but important case of the mechanics of a particle. Following Euler's example, Lagrange stated the prin ciple of the greatest or the least living force without Maupertuis. But Euler himself was determined to leave the honour of having to disco vered the principle of least action to Maupertuis ; and on this fact, he knew the evidence. The term u least " is only justifiable on the metaphysical plane, where every maximum would be evidence of the imperfection of the Creator's wisdom. Despite the criticism of Lagrange and, later, that of Hamilton, the name has survived and even now it is encountered in all the books. In the domain of the laws of impact Maupertuis' contribution was most constructive. His principle enabled him to encompass the cases of elastic bodies and hard bodies which had previously appeared separate, if not contradictory. A trace of this disjunction was still apparent in Lagrange's work. CHAPTER SIX EULER AND THE MECHANICS OF SOLID BODIES (1760) In 1760 Euler published a Theoria motus corporum solidorum seu rigidorum. This was eventually amended and added to by his son, in a new edition which appeared in 1790. The treatise starts with an introduction in which Euler confirms the principles of his Mechanica (1736) .x In connection with the mechanics of solids, Euler states that he will consider the characteristic property of a solid to be the conservation of the mutual distances of its elements. For every solid he defines a cen trum massae or centrum inertiae, remarking that the term " centre of grav ity " implies the more restricted concept of a solid that is only heavy, while the centre of mass of inertia is defined by means of the inertia alone (per solam inertiam determinari) , the forces to which the solid is subject being neglected. This apt comment has not prevailed against usage. Euler also defines the moments of inertia — a concept which Huyghens lacked and which considerably simplifies the language — and calculates these moments for homogeneous bodies. He systematically studies the motion of a solid body about a fixed axis, the given forces being at first zero and then being equated to the gravity alone. He demonstrates the existence of spontaneous or permanent axes of rotation for a solid body and thus clarifies the notion of the principle axes of inertia. He then investigates the motion of a free solid by decomposing it into the motion of the centre of inertia, and the motion of the solid about the centre of inertia. Euler clearly distinguishes — 1) the variation of the velocity of the centre of inertia I ; 2) the variation of the direction of the point I ; 3) the variation of the rotation of the solid about an axis passing through I. 4) the variation of this axis of rotation. 1 See above, p. 239. THE MECHANICS OF SOLID BODIES 277 We shall make this clear by an analysis of problems which Euler himself treated. " Problem 86. — Being given a solid body actuated by a given angular velocity about some axis passing through its centre of inertia, to find the elementary forces which must act on the elements of the solid in order that the axis of rotation and the angular velocity should undergo given variations in the time dt."1 Let I be the centre of inertia ; L4, IB, 1C the principal [or central] axes of the solid ; a, ft, y the angles between the axis of rotation and I A, IB, and 1C; co the angular velocity of rotation of the solid ; x, y, z the coordinates of some element of the solid with respect to the principal axes ; u, v9 w the components of the velocity of this element along the same axes ; X, Y, Z the unknown force applied to the particular element considered, whose mass is dM. The data of the problem are da, d/?, dy and doo and the unknowns, X, Y, Z. According to the fundamental law of Euler's dynamics, du, dv and dw are proportional to -r-r-r, -rvji and -^-. Therefore the problem reduces to the calculation of du, dv and dw. Now (u = co (z cos ft — y cos y) (dx = udt = a>dt (z cos /? — y cos y v = co (x cos y — z cos a) < dy = ---- w = CD (y cos a — x cos /?) ( dz = ---- A simple differentiation gives du = dco (z cos ]8 — y cos y) — wzdfi sin /? + coydy sin y + co2dt (y cos a cos /3 + % cos a cos y — x sin2 a) dv = ... dw = . . , and the unknown forces, X, Y, Z, applied to the element dM (x,y,z) are deduced from the fundamental law. In the next problem (No. 87) Eider calculates the moments P, (), jR, with respect to the principal axes, of the forces applied in the con ditions specified. By the definition of moments, dP = (ydw - zdv) dQ = . . . dR = . . . for the element dM. 1 Page 337 of the new latin edition of EULER'S -work, which we follow here. 278 THE PRINCIPLES OF CLASSICAL MECHANICS By summing over all the solid body, after replacing du, dv and dw by their appropriate value, Euler finds j P = — (Adco cos a — oAda. sin a -f- co2 (C — jB) at cos P cos y). /T?\ J dt (EMe= • •• > (A, JB, C, central moments of inertia of the solid.) Having obtained this result, Euler poses the following problem. " Problem 88. — If a solid body, turning about an axis passing through its centre of inertia with angular velocity o>, is acted upon by some forces, to find the variation of the axis of rotation and the angular velocity at the end of a time dt. *' A linear combination of the equations (E) gives the result in the form (C - B) (A - C) (B - A) da) = — — oral cos a cos p cos y ABC IP cos a Q cos /5 JR cos y 4- at I -\ h V A B C But Euler discovered that the equations can be cast into a much more simple form by introducing the components of the angular velocity of instantaneous rotation about the central axes of inertia ; that is, by introducing the quantities p = a) cos a, q — a> cos {3 and r = co cos y. Under these conditions a, /? and y no longer appear in the equations (E), which take the form Q = B + (A - C) rp Etder immediately saw the importance of these equations — " summa totius Theoriae motus corporum rigidorum his tribus formulis satis simplicibus continebitur. " Thus Euler's mathematical talent enabled him to discover the equations which express the general motion of a solid body under the influence of arbitrary forces by means of the decomposition of this motion into the motion of the centre of inertia and the rotation about the centre of inertia. An essential part of this process was the consideration of the central axes of inertia — that is, the con sideration of a real moving reference frame fixed in the solid. CHAPTER SEVEN CLAIRAUT AND THE FUNDAMENTAL LAW OF HYDROSTATICS 1. CLAIRAUT'S PRINCIPLE OF THE DUCT. Clairaut (1713-1765) was led to formulate the general law of the equilibrium of a fluid mass by the contemporary investigations of the figure of the Earth.1 Clairaut did not openly take sides in the conflict between the doctrine of vortices and that of attraction. However, he remarked that the Neo-Cartesians, while recognising part of the newtonian system, assumed a priori, whatever the shape of the Earth might be, that the gravity was inversely proportional to the square of the distance. They then compounded this gravitational force with the centrifugal force calculated for a given shape of the Earth. In the procedure of the attraction, on the other hand, the law of gravity depended on the shape of the Earth itself. " The Newtonians must find a spheroid such that a corpuscle, placed at an point on its surface and which is subject to both the centrifugal force and the attractions of all the parts of the spheroid, will take a direction perpendicular to that sur face. " 2 Huyghens assumed that the gravity must be normal at each point of the surface of a fluid mass. Newton supposed the equality of the weights of two liquid columns ending at the centre of mass. Bouguer had the merit of observing, as early as 1734 3, that these two hypotheses were incompatible for certain laws of gravity. Whence the theme of Clairaut's investigation — " To find the laws of hydrostatics which agree equally with all kinds of hypotheses about gravity. " 4 1 Theorie de la figure de la Terre tirie, des principes de Vhydrodynamique (Durand, Paris, 1743). 2 J/oc. eft., p. xxj. 3 Comparaison des deux lois que la Terre et les autres Planetes doivent observer dans la figure que la Pesanteur leur fait prendre, Mtimoires de FAcad&mie des Sciences, 1734. 4 CLAIRAUT, Joe. c£t., p. xxxij. 280 THE PRINCIPLES OF CLASSICAL MECHANICS Clairaut states the following principle at the beginning of his paper. "A mass of fluid cannot be in equilibrium unless the efforts of all the parts which are contained in a duct of any shape, which is imagined to traverse the whole mass, cancel each other out. " l 0 N He justifies this in the following way. " Since the whole mass PEpe is supposed to be in equilibrium, any part of the fluid could become solid without the remainder changing its condition. Suppose that all the mass is solidified except for what is necessary to form the duct ORS. The duct will therefore be in equilibrium. Now this can only occur if the efforts of 01? to leave the duct through S are equal to those of SR to leave through 0. " This principle includes Newton's hypothesis, as may be verified by the consideration of a duct MCN passing through the centre C. It also includes Huyghens' hypothesis — it suffices, indeed, to consider a duct FGD lying along the surface. This duct may be in equilibrium in two ways — the first, from the fact of Huyghens' hypothesis itself ; the second, because of the fact that a part FG thrusting towards D is compensated by a part GD thrusting towards F. But since the length of the duct is arbitrary, a small piece FG should be in equilibrium just as much as the whole duct, which excludes the preceding com pensation. Therefore it is necessary to return to Huyghens' hypothesis. But the most valuable form in which Clairaut stated his principle is the following one. 1 CLAIRAUT, Joe. tit., p. 1. CLAIRAUT AND HYDROSTATICS 281 " In order that a mass of fluid may be in equilibrium^ it is necessary that the efforts of all the parts of the fluid which are contained in a duct which is re-entrant upon itself should cancel each other out. " a This proposition can be justified immediately by solidifying all the fluid except that in the duct IKLT. The efforts of the parts IKL, ITL must be equivalent to each other, " or else there would be a perpetual current in the duct. " It can also be deduced from the preceding proposition by the consideration of two ducts FIKLG andFITLG. Fig. 94 Clairaut then observes that in the consideration of two ducts ab^ a/?, which are filled with liquid and rotate about an axis Pp, the total effort of the centrifugal force on the duct ab will be the same as that 1 CLAIRAUT, Joe. cit., p. 5. 282 THE PRINCIPLES OF CLASSICAL MECHANICS on the duct a/? if a and a, b and /5 are respectively at the same distance from the axis. It follows that " when it is desired to investigate whether a law of gravity is such that a mass of fluid turning about an axis can preserve a constant shape, no purpose it served by paying attention to the centrifugal force. That is, that if the mass of fluid can have a constant shape when not rotating, it will also be able to have one when rotating. " If a duct abed is considered in the mass, this must be in equilibrium in order that the mass should have a constant shape. Now it is seen that the sum of the effects of the centrifugal force on abed is nothing ; for ab and cb will thrust on b to the same extent, just as ad and cd will thrust equally on d . Therefore the rotation will not prevent the equilibrium of a duct which is re-entrant upon itself. Accordingly, " if the duct is in equilibrium when only gravity alone is considered, it will still be in equilibrium if it is supposed that, instead of gravity, the actual weight, composed of gravity and the centrifugal force, is considered. " 2. THE CONDITION TO BE SATISFIED BY THE LAW OF GRAVITY TO ASSURE THE CONSERVATION OF THE SHAPE OF A ROTATING FLUID MASS. Clairaut supposes that gravity has two components, P and Q, which are parallel to the axes CP and CE. He considers an arbitrary duct ON which ends on the surface, and an element of this duct, Ss9 which has Sr = dx and sr = dy. CLAIRAUT AND HYDROSTATICS 283 The force Q acts along SH. The projection of Q on the direction rs of the duct has the value Q • ^. By multiplying by the mass of the oS element, it is found that " the effort which the force Q will cause the cylinder to exert on the point 0" has the value Qdy. Similarly, the force P gives the effort Pdx. Whence the total effort of gravity is N It is necessary that the sum of the efforts of gravity on any duct ON should have the same value as if any other duct passing through the points 0, N had been taken. " The equilibrium of the fluid requires that the weight of ON does not depend on the curvature of OSN or, that is, on the value of y as a function of x. Therefore it is necessary that Pdx + Qdy can be integrated without knowing the value of x ; that is, it is necessary that Pdx -f- Qdy should be a complete differential, or that . -r= LJL. » 1 dy dx 3. CONDITION FOR THE EQUILIBRIUM OF A FLUID MASS. Clairaut set himself [the following problem.2 " Supposing that the force which actuates the particles of a fluid had been decomposed into three others, of which the first acts perpendicularly to the plane 1 CLAIRAUT gave this last condition in the Memoires de I* Academic des Sciences for 1740, p. 294. 2 CLAIRAUT, loc. cit., p. 96. 284 THE PRINCIPLES OF CLASSICAL MECHANICS QAP and the second and third along two perpendicular directions, QA and AP, in the plane QAP, it is required to find the relation there must be between these three forces in order that the equilibrium of the fluid may be possible. " Fig. 97 If P, Q and R are the components of the force parallel to AP, AQ and AR, and Nr9 rs and $n are represented by dx, dy and dz, then an argument quite analogous to that of the preceding § gives the condition Pdx _j_ Qfy -|_ Rdz = complete differential or 'By Hz __ 3z~~ dy* Apart from this, Clairaut also defines the lines and surfaces of intensity of gravity. He verifies that a fluid mass that is imagined to be divided into an infinite number of layers which are defined by the surfaces of intensity, will be in equilibrium if, at each point of one of the surfaces, the weight is inversely proportional to the thickness of the layer. This remarkable analysis, which can be regarded as the introduction of the concept of potential, enabled Clairaut to make an important contribution to the theory of the figure of the Earth. Newton had assumed, a priori, the shape of an elliptic spheroid. He considered two columns — one connecting the centre to a Pole and the other, the centre to the Equator — and equated the difference in their weight to the sum of the centrifugal force on the different portions of the Equitorial 229 column. The ratio of the axes obtained in this way was TTTTT:- The Neo-Cartesians, on their side, announced the ratio "r^r. CLAIRAUT AND HYDROSTATICS 285 As early as 1737 Clairaut was able to show that the elliptic spheroid was an equilibrium figure. 230 Newton's value — or that of MacLaurin ;rry — supposed the Earth to be homogeneous. This value was disproved by experiments made in Lapland by an expedition sent at the command of the King. This expedition consisted of four members of the Academic des Sciences — Clairaut, Camus, Le Mourner and Maupertuis — who were joined by two others — Celsius and the Abbe Outhier. It embarked at Dunkirk on May 2nd, 1736.1 The relative magnitudes of degrees of meridian obtained in this way indicated a flattening about ^-r, less than that which Newton had announced. It was therefore necessary to give up the hypothesis of the homogeneity of the Earth. If the Earth were formed of similar layers, it shape would not conform to the fundamental law of hydro statics. And Clairaut decided on the existence of layers which were flatter as they were further from the centre, the flattening following a law that depended on the decrease of the density between the centre and the surface. ^ l For the account of this mission, see MAUPERTUIS' Discours lit dans VAssembtie publique de V Academic Royale des Sciences sur la mesure de la Terre au Cercle polaire (CEuvres completes, Vol. Ill, 1756, p. 89). CHAPTER EIGHT DANIEL BERNOULLI'S HYDRODYNAMICS D'ALEMBERT AND THE RESISTANCE OF FLUIDS EULER'S HYDRODYNAMICAL EQUATIONS BORDA AND THE LOSSES OF KINETIC ENERGY IN FLUIDS 1. RETURN TO THE HYDRODYNAMICS OF THE xviith CENTURY. We have already described the attempts of Newton and Varignon to explain the law which Torricelli had formulated. We must now return to the work of Mariotte, who emerges as the forerunner of the important XVIIIth Century school of hydrodynamics. As early as 1668 a Committee of the Academic des Sciences was formed and instructed to verify TorriceUi's law experimentally. The Committee's members were Huyghens, Picard, Mariotte and Cassini. It extended its investigations to the determination of the effect of the impact of a fluid stream on a plane surface. Influenced by these investigations, Mariotte published, in 1684, the Traite du mouvement des eaux in which he carried the subject further. He verified Torricelli's law without observing the contraction of the stream. Newton corrected this error in the second edition of the Principia. In the matter of the impact of a fluid stream on a surface, Mariotte had the merit of demonstrating the importance of the deviation from the momentum of the fluid. But he compared this problem with the action of a fluid current on a completely immersed solid, thus disregarding the reconstitution of the stream-lines behind the obstacle. Mariotte was also the first to introduce considerations of similitude in the resistance of fluids, and the first after Huyghens to state that the resistance of a fluid was proportional to the square of the velocity. Finally, Mariotte turned his attention to hydraulics, and studied the velocity of flow in rivers or canals, and the friction of water in pipes. HYDRODYNAMICS 287 2. DANIEL BERNOULLI'S HYDRODYNAMICS. The beginning of the XVIIIth Century was a period of extraordinary development for both theoretical and applied hydrodynamics. In the compass of the present history — limited to a study of the evolution of the principles of dynamics — it would be impossible to analyse the complex development of the mechanics of fluids in any detail. This investigation rapidly became an independent branch of science, both theoretical and experimental. However, we consider it valuable to deal with a few typical achie vements of the XVIIIth Century in this field. In 1738 Daniel Bernoulli published Hydrodynamica, sive de viribus et motibus fluidorum commentarii. This was a most remarkable work which has only aged a little in the last two centuries. In D, Bernoulli's sense, the term hydrodynamica includes hydro statics — the science of equilibrium — and hydraulics — the science of fluid motion. " My theory is novel, " he declared, ** because it considers both the pressure and the motion of fluids. It might be called hydraulico- stattica. " D. Bernoulli's guiding principle was that of the conservation of living forces or, more accurately, that of the equality between the actual descent and the potential ascent (aequalitas inter descensum actualem ascensumque potentialem) . As the controversy on living forces was then at its height, D. Bernoulli took certain precautions in this respect. Quite legiti mately, he emphasised that the doctrine of Leibniz stemmed, in fact, from a principle that Huyghens had formulated. (No body freely can rise to a height greater than that from which it has fallen.) * Moreover inelastic impact, which entails a loss of kinetic energy, has its analogy in hydrodynamics, where it appears as a reduction of the ascensus potentialis. D. Bernoulli excluded this occurrence from the theory, adding that this was a reason for applying the theory with care. Apart from the hypothesis of the conservation of living forces, D. Bernoulli also assumed that all the particles of a slice of fluid which was perpendicular to the direction of motion moved with the same velocity, which was inversely proportional to the cross-section of the slice. Further, he only studied stationary states (fingenda est unifor- mitas in motu aquarum). We shall give a concrete example of one of the numerous problems which D. Bernoulli solved. 1 See above, p. 187. 288 THE PRINCIPLES OF CLASSICAL MECHANICS "Let there be a vessel of very large cross- section, ACEB, which is kept full of fluid, and let it be pierced with a horizontal cylindrical tube ED. At the end of the tube is an orifice 0 through which the fluid escapes with constant velocity. It is required to find the pressure exerted on the walls of ED. B Fig. 98 a c r I I 6 b d D " Let a be the height of the surface AB above the orifice o. When the steady state is established (siprimafluxus momenta excipias) the velo city with which the water leaves through o is constant and equal to -y/ a, since we suppose that the vessel remains full. If n is the ratio of the section of the tube to that of the orifice, the velocity of the water in / — the tube will be — — . If the whole end FD were missing the velocity 71 of the water in the tube would be \/a, which is greater than Therefore the water in the tube strives towards a greater motion, 71 to which the end FD presents an obstacle. From this there results an over-pressure which is transmitted to the boundary walls. The press ure on the boundary walls is thus proportional to the acceleration which the fluid would take if the obstacle were instantaneously taken away and the fluid were allowed to escape into the atmosphere. " All this happens as if, during the escape through the orifice o, the tube FD were suddenly broken off at cd and the acceleration of the small portion of fluid abed were sought. . . . Thus we must consider the vessel A BEcdC and, with its help, try to find the acceleration A/ ^ which the particle, of velocity — — , takes on escaping. HYDRODYNAMICS 289 ** Let v be the variable velocity of the water in the tube Ed ; n be the cross-section of the tube ; c its length, equal to EC ; and let dx be the length ac. A portion of the fluid enters the tube at E at the same time that abed escapes from it. The portion at E, whose mass is ndx, acquires the velocity v or the living force nv2dx, which is generated suddenly. Indeed, since the section of the vessel Ae is infinite, the portion of the liquid at E does not have any velocity before entering the tube. To the living force nvzdx will be added the increase of the living force which the water receives in Eb when the portion ad escapes — say 2ncvdv. These two quantities together are due to the real descent of the portion of the fluid from the height BE, or a. Therefore 2ncvdv = nadx or vdv a — dx 2c " Throughout the motion the increase dv of the velocity is propor tional to the pressure produced in the time — . Therefore the press ure which is exerted on the portion ad is proportional to the quantity dv , . a — t;2 v — ; that is, to dx ' ---- ' 2c At the instant when the tube is broken, v = — — or v2 = — -. n n2 71 ~ - 1 i — This expression is to be substituted in — - - , which becomes a ^ 2c " And it is this quantity which is proportional to the pressure of the water on the portion ac of the tube, whatever the section of the tube may be and whatever the orifice in the end might be. ... " If the orifice is infinitely small, or n is infinitely large compared with unity, it is clear that the water exerts the whole pressure cor responding to the height a. This pressure we call a. But, then, unity is vanishingly small compared with n2 and the quantity to which the pressure is proportional has the value — . . . . If the quantity — - £c £>c corresponds to the pressure a, the pressure corresponding to the quantity n2 _ i n2 _ i a - - — will be a - - — , which is independent of c, Q.E.D. " 2n2c JIT There is no need to quote further from among the demonstrations which D. Bernoulli developed in a similar way. They are remarkable for their ingenuity and all start from the single hypothesis which we have 19 290 THE PRINCIPLES OF CLASSICAL MECHANICS recorded. We notice that in the same treatise D. Bernoulli enunciated the theorem with which his name is still associated, and which appears, in a different form, in all the classical treatises on hydrodynamics. D. Bernoulli also treated the impact of a fluid stream on a plane in a way which was superior to that used hy Mariotte. He showed that this problem was distinct from the one which concerns the effect of a fluid current on a completely immersed solid. Together with the theoretical solutions of the problems which he treated, D. Bernoulli recorded much experi mental material in support of his demonstrations. We also remark, without being able to devote a discussion to it, the fact that MacLaurin (in his Treatise on Fluxions) and Jean Bernoulli (in his Nouvette hydraulique) had sought to dispense with the assumption of the conservation of living forces which D. Bernoulli had made. But, in the words of Lagrange, " M acLaurin's theory is not very rigorous and seems to be contrived in advance to agree with the results that he wished to obtain. " As for that of Jean Bernoulli, it leaves much to be desired in clarity and accuracy and, in d'Alembert's opinion, " its general principle is deduced so easily from that of the conservation of living forces that it appears to be nothing else than that same prin ciple presented in another form ; and again, Jean Bernoulli seeks to confirm his method by means of indirect solutions supported by the laws of the conservation of living forces. " 3. D'ALEMBERT AND THE MOTION OF FLUIDS. D'Alembert's contribution to the theory of fluids was both extensive and important. In the confines of this history we must restrict our selves to the indication of the principles of this work, and to the citation of some typical examples. Generally speaking d'Alembert remained faithful, in his treatment of the mechanics of fluids, to the Discours Preliminaire of his Traite de Dynamique. Nevertheless he did not carry into this field the convic tion that science could be of a purely rational origin. He wrote, " The mechanics of solid bodies depends only on meta physical principles which are independent of experiment. Those principles which must be used as the foundation for others can be determined exactly. The foundation of the theory of fluids, on the other hand, must be experiment, from which we receive only a very little enlightenment. " After having devoted long and laborious efforts to an attempt at elucidating the motion and resistance of fluids, d'Alembert remained optimistic about the future of this theory. " When I speak of the HYDRODYNAMICS 291 limitations by which the theory must be prescribed, " he wrote, " I only contemplate the theory with such assistance as it can now obtain, and not the theory as it may be in the future, aided by what is yet to be discovered. For, in whatever subject one might be, one should not be too ready to erect a wall of separation between nature and the human mind. " 1 However, d'Alembert did not conceal the great difficulties that arose in the translation of such complicated phenomena as those of the motion and the resistance of fluids into a rational language. In this period of enlightenment, when all men, including d'Alembert, were willing to trust — perhaps too readily — in the universal validity of science, he made the reservation — " not to exalt the algebraic formulae into physical truths or propositions too readily. . . . Perhaps the spirit of calculation which has displaced the spirit of system rules, in its turn, a little too strongly. For in each century there is a dominant style of philosophy. This style almost always entails some prejudice, and the best philosophy is the one which has least of this consequence. " 2 In 1744, one year after the publication of his Traite de Dynamique, d'Alembert wrote a Traite de FEquilibre et du Mouvement des Fluides. In it he refused to start from the principle of the conservation of living forces, as Daniel Bernoulli had done in his Hydrodynamica. For this principle, as we know from the discussion in the Discours Preliminaire, d'Alembert regarded as not being sufficiently well-established to be used at the basis of hydrodynamics. " One of the greatest advantages that follow from our theory is that of being able to show that the well- known law of mechanics called the conservation of living forces is as appropriate in the motion of fluids as in that of solid bodies. " D'Alembert, a keen critic of his predecessors, reproached Daniel Bernoulli for not having brought forward, in Volume II of the Memoires de Petersbourg (1727), " other evidence for the conservation of living forces in the fluids than that a fluid could be regarded as aggregate of fluid particles which press on each other, and that the conservation of living forces is generally accepted to be applicable to the impact of a system of bodies of this kind. . , . Therefore it seemed to me that it is necessary to prove, more clearly and exactly, the question of whether the principle is applicable to fluids. I had tried to demonstrate this, in a few words at the end of my Traite de Dynamique.* Here will be found a more detailed and more extended proof. " 4 1 Essai d'une nouvelle thforie de la resistance des fluides (David, Paris, 1752), p. xxxiv. 2 Ibid., p. xlj. 3 Traite de Dynamique (1758 ed.), p. 269. 4 Trait^ de VEquilibre et du Mouvement des Fluides (1744), Preface. 292 THE PRINCIPLES OF CLASSICAL MECHANICS At the beginning of the Traite des Fluides d'AIembert explicitly stated the hypothesis of flow by parallel slices of fluid, whose parallelism was conserved throughout the motion. Moreover, he assumed that all the points of the same slice had the same velocity. Necessarily this hypothesis restricted the generality of his analysis. Given these hypotheses, d'AIembert extended to fluids the principle that he had used as a basis for his dynamics. " In general, let the velocities of the different slices of the fluid, at the same instant, be represented by the variable v. Imagine that dv is the increment of the velocity in the next instant, the quantities dv being different for the different slices, positive for some and negative for others. Or, briefly, imagine that v + dv expresses the velocity of each layer when it takes the place of that which is immediately below. I say that if each layer is supposed to tend to move with an infinitely small velocity + dv, the fluid will remain in equilibrium. " For since v = v + dv + dv, and the velocity of each slice is supposed not to change in direction, each layer can be regarded, at the instant that v changes to v +. dv, as if it had both the velocity v + dv and the velocity + dv. Now, since it only retains the first of these velocities, it follows that the velocity + dv must be such that it does not affect the first and is reduced to nothing. Therefore if each slice were actuated by the velocity + dv the fluid would remain at rest. " 1 D'AIembert accompanied this theorem with the following observation. " If it is supposed that the particles of the fluid are subject to an accelerating force 9?, different for each slice if so desired, then it is clear that at the end of the instant dt the velocity v will be v + cpdt if the slices do not interact in any way. Therefore, if at the end of the instant dt the velocity v becomes v If dv because of the interaction of the slices, it would be necessary to suppose that v + cpdt = v + d v + (pdt + dv and it is clear that the fluid would remain in equilibrium if only the slice were actuated with the velocity <pdt + dv. " 2 Starting from this principle, d'AIembert went back to the problems which had been studied by Daniel Bernoulli and treated them anew. We shall pause on a single example, the first and the most simple which d'AIembert treated. The question is that of the flow in parallel slices of a fluid which 1 Traite de rEquilibre et du Mouvement des Fluides (1744), p.70. 2 Ibid., p. 71. HYDRODYNAMICS 293 is supposed incompressible in a vessel of any shape. The licpiid is homogeneous, has no weight, and is set in motion by an unspecified means ; for example, by the impulsion of a piston. Let u be the velocity of the layer GH. That of the layer CD will , uGH beW If y is the width, and x the side, of one of the slices, then it is possible to write ydx = constant. B b Fig. 99 Between the instant t and the instant t + dt the fluid moves from the position CDLP to the position cdlp. If v is the variable which represents the velocity of a slice in the position CDLP and if v — dv is (algebraically) the value of the same variable at cdlp? the fluid will remain in equilibrium if each slice tends to move with the velocity dv. D'Alembert expresses this by J dvdx = 0 294 THE PRINCIPLES OP CLASSICAL MECHANICS or f ydx • dv n uGH But v = . Therefore y f ydx*vdv _ J ~H~ = °* Now " GH is constant as well as «, with respect to the variables v and dv " and ydx is constant. It follows therefore that fyfa.V* = fydx-v*. Here V is the velocity of each slice in the instant which follows that at which its velocity is v. " Therefore it is seen that the principle of conservation of living forces also applies to fluids. " To (TAlembert however, this principle was a corollary which had to be verified in each instance. And indeed, he was not surprised to retrieve, by means of his principle, the solutions of the problems which had been treated by D. Bernoulli. In this connection, d'Alembert makes the following observation in the preface to his Traite des Fluides. " And indeed I am forced to confess that the results of my solutions always agree with those of M. Daniel Bernoulli. Nevertheless it is necessary to except a small number of these problems. These are problems in which that skilful geometer used the principle of the conser vation of living forces to determine the motion of a fluid in which there is a portion whose velocity increases suddenly by a finite quantity. Such, for example, is the problem in which the question is to find the velocity of a fluid leaving a vessel which is kept filled to the same height. It is supposed that the small sheet of fluid which is added at each instant receives its motion from the fluid below, by which it is drawn along. It is clear that in a similar hypothesis this sheet of fluid, which had no velocity at the instant that is was added to the surface, in the next instant receives a finite velocity equal to that of the surface which draws it along. Now, without wishing to ask whether or not this hypothesis is in conformity with nature, it is certain that the principle of living forces should not be used to investigate the motion of any system when it is supposed that there is some body in this system whose velocity varies in an instant by a finite quantity. " The Traite des Fluides also contains an attempt to investigate the resistance of fluids. D'Alembert used rather a flimsy mechanical model, analogous to that which he had reproached Daniel Bernoulli for using, in order to substantiate the conservation of living forces. HYDRODYNAMICS 295 " First I determined the motion that a solid body must communicate to an infinity of small balls with which it was supposed to be covered. Then I showed that the motion lost by this body in a given time was the same whether it collided with a certain number of balls at once, or whether it merely collided with them in succession. Further, I showed that the resistance would be the same when the corpuscles had some other shape than the spherical one, and when they were arranged in any manner that might be desired, provided that the total mass of these small bodies which was contained in a given space remained the same. By this means I arrived at the general formulae for the resistance, in which there only appeared the relationships of the den sities of the fluid and the body moving in it. " x Finally, d'Alembert devoted a chapter to " fluids that move in vortices and to the motion of bodies which are immersed in them. " He did not make this study to bolster up " a cause as desperate as that of Descartes' vortices, " but because the subject seemed in itself to be " rather curious, independently of any application to the case of planets that one might desire to make. " 4. D'ALEMBERT AND THE RESISTANCE or FLUIDS — His PARADOX. The Essai d'une nouvelle theorie de la resistance des fluides (Paris, David, 1752) has its origin in d'Alembert's participation in the competi tion held by the Academy of Berlin in 1750. The subject chosen concerned the theory of the resistance of fluids. The prize was not awarded, the Academy having required the authors to give proof of the agreement of their calculations with experiment by means of a supplement to their work. D'Alembert, who had competed, seems to have become rather bitter at this decision, for the observations available at that time were often contradictory and the theory was sufficiently difficult to require a worker's undivided attention. His Essai is, apart from some details, his contribution to the Berlin competition. In his Essai d'Alembert obtains, at least for plane motions, the general equations of the motion of fluids.2 However, his analysis is so long and tortuous that we cannot attempt to summarise it. In this achievement d'Alembert preceded Euler by some years — but he did not succeed in presenting the equations of hydrodynamics in the " direct and luminous " 3 way that Euler was able to discover. 1 Traite de V£quilibre et du Mouvement des Fluides, Preface. 2 See also Opuscules mathematiques by d'ALEMBERT, especially Vol. VI, p. 379 (1778 ed.). 8 LAGRANGE, Mecanique analytique. Part II, Section X. 296 THE PRINCIPLES OF CLASSICAL MECHANICS Faithful to his principles, cPAlembert reduced the search for the resis tance to the laws of equilibrium between the fluid and the body. The resistance was given by the momentum lost by the fluid. Rather than attempt to analyse the difficult Essai sur la resistance des fluides? we consider it more valuable to reproduce a portion of the work in which d'Alembert, guided, it seems, by his logical insight alone, discovers the hydrodynamical paradox with which his name is still associated. " Paradoxe propose aux geometres sur la resistance des fluides.1 66 Suppose that a body is composed of four equal and similar parts and placed in an indefinite fluid contained in a rectilinear vessel. " Imagine that the body is fixed and immoveable, and that the parts of the fluid all receive an equal impulsion parallel to the sides of the vessel and the axis of the body. It is clear that the parts of the fluid at the front of the body must be deflected and must slide along the body, thus describing curves which are more like straight lines as they are further from the body ; this up to a certain distance, which will be at least that of the boundary walls of the vessel, where the parts of the fluid will move in straight lines. " If the solid is not terminated in a very sharp point, in such a dy way that the derivative -f- may be either finite or infinite at the dx origin, there will be or there may be a small portion of the liquid which is stagnant at the front of the body.2 But in order to avoid this diffi- 1 Opuscules mathematiques. Vol. V, p. 132. 2 We reproduce here the diagram used by d'ALEM- BERT. The " stagnant part, " which, in d'ALEMBERx's opinion, -will exist in general, is referred to in the Essai sur la resistance des fluides, in the following terms — " All moving bodies which change direction only do so by imperceptible degrees. The particles which move along TF do not travel as far as A because of the right angle TA& — they leave TF at F, for example. There fore, in front of and behind the solid, there are spaces in which the fluid is necessarily stagnant. " Fig. 100 HYDRODYNAMICS 297 c?v culty, I suppose that -j- = 0 at the origin, so that the point of the body u/x is infinitely sharp. Then there will no longer be stagnant fluid, and the fluid will run along the forward surface as far as the point at which dy again becomes parallel to the axis. 44 With regard to the backward part, it seems at first sight that the motion must be different from that at the forward part. For at the forward part the fluid is not free to follow its original direction, while it is so free at the backward part. However, for the moment suppose that the particles of the fluid do have the same motion at this backward part as at the front. It is easily discovered, by our theory of the motion of fluids, that under this supposition the laws of the equilibrium and the incompressibility of the fluid will be fully satisfied. For since the back is similar to the front (hyp.), it is easy to see that the same values of p and q I that will yield the equilibrium and the incompressibility of the forward part will yield the same results for the backward part. Therefore there is nothing but justification for the supposition concerned. Therefore the fluid can move in this way at the back. Now if it can, it must. For there is only one possible way in which the fluid can be moved by the body. " In this condition, the fluid will exert a pressure on the body, though this pressure will not have the effect of separating the body from its position because the body is immoveable and fixed at rest in the middle of the fluid (hyp.). Let u be the velocity imparted to the fluid. The pressure, if it exists at the first instant, will be exerted on the front of the body and will be &u, where k is a quantity which depends on the shape of the body. 44 Let uq be the velocity of the fluid parallel to the axis and let up be the velocity of the fluid perpendicular to the axis. Since it is supposed that, at the first instant, with velocity u parallel to the axis and that it changes this velocity into the velocities uq and wp, the fluid will exert a pressure on the body which will be the same as if the fluid were at rest and the body were moved with velocities u — uq and u—up. Now because of the velocity w, the pressure exerted will be, by the principles of hydrostatics, equal to Mu and in the opposite direction to u, if M is the mass of the body. And because of the velocities — uq and — up, the pressure exerted will be in a direction opposite to that of Mu and will be equal to 4u / dy J ds \/(p* + g2)- It is easy to see this by supposing that / ds *\/(p2 + q2) — 0 and J dy J ds \/(p* + q2) = 0 at the point at which dy = 0 and which 1 Quantities proportional to the components of the velocity in d'ALBMBERT's theory. 298 THE PRINCIPLES OF CLASSICAL MECHANICS is not the summit of the body — that is, at the point at which the tangent is parallel to the axis. For it is clear that, since dy = 0 (hyp.) for x = 0 and the body is composed of four equal and similar parts, there must be a point on each side of the axis, and in the centre of the axis, where dy = 0. Therefore, if it is supposed that J" dy J ds \/(p2 -f- g2) = J?, in its totality, then k = 4R — M. " In the following instants parts of the fluid evidently retain the velocities uq and up, from which it is easy to see, by means of our theory on the resistance, that the pressure of the fluid on the body will be absolutely nothing. For the pressure on the forward surface is equal and opposite to the pressure on the backward surface. *4 And if it were supposed that a force y, constant or variable from one instant to another but always the same for all parts of the fluid at the same instant, acted on all these parts, they would nevertheless continue to describe the same lines with a velocity that would be increased in the ratio of ydz to u. And from this it would only result that a new pressure equal to ky was exerted on the surface of the body. " Suppose now that the fluid is at rest and the body is moved along in it with the velocity v, of which it only retains a part u. Give the whole system — fluid and body — a motion u in the opposite direction. The body will be at rest in fixed space. In front there will be the force M (v — u), while the fluid will exert a pressure ku on the body. The latter will nullify the former. " Therefore 7 / \ n/r Mv Mv ku = (v — u) M u — = k + M 4B " Now suppose that the body continues to move in the fluid with a velocity which decreases by an amount ydt at each instant so that du = — ydt. Also suppose that the system of body and fluid moves on the opposite direction with this decreasing velocity. It is apparent that the body will be at rest and that the pressure at each instant will be ky, or kdu, which must counterbalance Mdu. Therefore Mdu = du (4H — M) or Mdu == 2Rdu. " Therefore either du == 0 or 2R = M. " If du = 0, the body will move uniformly and it will be true Mv that u = — . " If 2R = M, then u = - and the quantity du remains indeter minate. It could be supposed to be zero, and even must be supposed to HYDRODYNAMICS 299 be zero, since there is no other equation to determine it than Mdu = 2Rdu. " Thus the greatest alteration that could occur in the original mo tion of the body is that the velocity v — which is supposed to have been impressed on it — should be changed to -— in the first instant, and after this the body will move without suffering any resistance due to the fluid. _ " If the shape of the body is such that J dy J ds \/(p* + q*) = R= — then u = v. Whence Mdu = du (4J? — M) = 0 and therefore the body will not lose any velocity in the first instant. This seems also borne out by experiment. 46 I do not ask whether the quantities p and q which are obtained by the theory are such that 4>R = M for any shape of the body — it appears rather doubtful that this should be so. Neither do I ask whether 4J? could be greater than M for some shapes and less than M for others. These conditions would imply u < v (contrary to exper iment) and u > v (contrary to common sense). Therefore we will have du = 0 and, from our theory, it will follow that the body, supposed to be of four equal and similar parts, will suffer no resistance from the fluid. " And whatever relation there is supposed to be between 4.R and M, it is apparent that the velocity v will, at the most, only experience an alteration in the first instant, and that it will then remain uniform. This would be much worse if 4sR < M for then the initial velocity would first increase and afterwards remain uniform. " I must therefore confess that I do not know how the resistance of fluids can be explained by the theory in a satisfactory way. On the contrary, it seems to me that this theory, handled with all possible rigour, yields a resistance which is absolutely nothing in at least several situations. I bequeath this strange paradox to the geometers, that they may explain it. " 5. EULER AND THE EQUILIBRIUM OF FLUIDS. In a paper given to the Academy of Berlin in 1755,1 Euler directed his attention to the equilibrium of fluids. He considered a fluid, either compressible or not, which was subjected to any given forces. " The generality that I include ," he declared, " instead of dazzling us, will rather discover the true laws of Nature 1 Principes g£neraux de Fetat tfequilibre des fluides, Memoires de V Academic de Berlin, 1755, p. 217. 300 THE PRINCIPLES OF CLASSICAL MECHANICS in all their splendour, and there will be found yet stronger reasons for wondering at their beauty and simplicity. " The general problem which Euler poses is the following one. " The forces which act on all the elements of the fluid being given, together with the relation which exists at each point between the density and the elasticity of the fluid, to find the pressures that there must be, at all points of the fluid mass, in order that it may remain in equilibrium. " In the fluid mass Euler considers an elementary rectangular parallel- ipiped with one corner at the point Z, of coordinates x, y, z and with sides dx, dy and dz. The components of the " accelerative force " applied to each element are called P, Q and R, and q is the density of fluid at Z. Then the element of volume dxdydz is subject to the " motive force " whose components are Pqdxdydz Qqdxdydz Rqdxdydz. If p is the unknown pressure at the point Z, then dp = Ldx + Mdy + Ndz. By a very simple geometrical argument — which has become class ical — Euler deduces the general conditions of equilibrium L = Pq M = Qp N = Rq. If L, M and N are the partial derivatives of a function p (x, y, z), equilibrium requires the conditions d (Pq) = d (Qq) d_(Qq^^d^q) d (Rq) = d (Pq) dy dx dz dy dx dz If p is a given function of q at each point of the fluid, the relation dp = q (Pdx + Q dy + Rdz) shows that Pdx + Qdy + Rdz is the total differential of the func- . dp tion — . q This differential represents the " effort " or the " efficacy " of the given force — this was a notion which Euler used in the case of central forces. In a second paper, which will be discussed in the next §, Euler deduced a general conclusion from the equation of equilibrium ^ = Pdx + Qdy + Rdz. HYDRODYNAMICS 301 ** The forces P, (), R must be such that the differential form Pdx -f Qdy + Rdz either becomes integrable when the density q is constant or uniquely dependent on the elasticity p, or becomes integrable when multiplied by some function. " Euler did not refer to Clairaut in this connection. Although he had the merit of introducing the pressure and relating it to the acceler- ative force at each point it must be observed, with Lagrange, that Euler 's achievement was that of applying, by generalising it, the principle of Clairaut. 6. EULER AND THE GENERAL EQUATIONS OF HYDRODYNAMICS. We now come to a fundamental paper of Euler on the equations of hydrodynamics.1 So perfect is this paper that not a line has aged. In assuming this difficult task, Euler declared, " I hope to emerge successful at the end, so that if difficulties remain they will not be in the field of mechanics, but entirely in the field of analysis. " Euler considers a fluid which is compressible or incompressible, homogeneous or inhomogeneous. Its original state — that is, the arrange ment of the particles and their velocities — is supposed known at a given instant, as are the external forces acting on the fluid. It is necessary to determine, at all times, the pressure at each point of the fluid together with the density and the velocity of the element passing through that point. In order to study the present state of the fluid, Euler uses the com ponents of the accelerative force, P, Q, R which are known functions of #, y, z and Z.2 The density q, the pressure p, and the components u, v9 w of the velocity of the element of the fluid which is at the point Z at the time * are unknown. During the time dt the element of fluid at Z will be carried to the point Z', whose coordinates will be x + udt y + vdt z The element of fluid at 2, of coordinates x + dx y + dy z + dz 1 Principes g$n$raux du mouvement des fluides, Memoires de VAcademie de Berlin, 1755, p. 274. 2 EULER also refers to a variable r, the " heat at the point Z, or that other property which, apart from the density, affects the elasticity. " 302 THE PRINCIPLES OF CLASSICAL MECHANICS has a velocity whose components are . du , du _ du , u + — dx + — dy + -5- dz dx dy dz v+ ... w + and, during the time df, is carried to the point z'. In order to perform the calculation, Eider first considers a segment Zz which is parallel to the axis of x. During the time dt this segment will turn through an infinitely small angle, and its length will become to the second order. In a latin paper, Principia motus fluidorum, Euler elucidates the problem — then entirely novel — of the kinematics of continuous media. He calculates the form which the elementary parallelipiped, whose origin is Z and sides are dx, dy and dz, will assume at the time t + dt because of the motion of the fluid. He finds that the volume becomes 7 , , /, , , du , T dv , , dw\ dxayaz 1 1 + at - — \- dt - — h dt -r- • \ dx dy dz/ Similarly the density, g, of the fluid at Z becomes, at Zr, . ot dx dy dz At this point Euler expresse the conservation of the mass in the course of the motion. " As the density is reciprocally proportional to the volume, the quantity q' will be related to q as dxdydz is related to j j j /i i 7 du . - dv , . dw\ dxdydz 1 + dt — + dt — + dt — ; \ ox dy dzj whence, by carrying out the division, the very remarkable condition which results from the continuity of the fluid, dq , dq dq dq du , dv dw s-t + ud-x + v^ + wr2+^x + ^ + ^ = °' This may be written more simply as HYDRODYNAMICS 303 and, for an incompressible fluid, it reduces to aufoSw; „ dx~^ dy~^ dz Euler then calculates the acceleration of the element of fluid which is at Z at the instant t. He first writes the components of the velocity at the point Z', to which the point Z is carried at the end of the time dz, in the following form . 7 du , 7 du , u + at - — \- udt - — k dt dx v + ... du , 7 du - — \- wdt — dy dz whence the acceleration or the increment of the velocity du , Su , , ^ — r w -^~ dy ^ dz 3v St dw The pressure exerts the " accelerative force, " whose components are 1 dp q dx 1 dp q dy I dp q dz on the elementary mass of the parallelipiped. Thus the equations of motion of the fluid, to be joined to the equation of continuity, are du , du ^ 1 dp du , du p -- ^- = - -- \- u- -- - q dx dt ' dx ' dy ^r- dz Euler was too aware to misunderstand the difficulty of the study of these equations of motion. Thus he wrote — " If it does not allow us to penetrate to a complete knowledge of the motion of fluids, the reason for this must not be attributed to mechanics and the inadequacy of the known principles, for analysis itself deserts us here. ..." Lagrange, in this connection, wrote — 304 THE PRINCIPLES OF CLASSICAL MECHANICS " By the discovery of Euler the whole mechanics of fluids was reduced to a matter of analysis alone, and if the equations which contain it were integrable, in all cases the circumstances of the motion and behaviour of a fluid moved by any forces could be determined. Unfortunately, they are so difficult that, up to the present, it has only been possible to succeed in very special cases. " Without concerning ourselves with the particular problems which he treats, we remark that Euler indicated the simplicity that results if udx + vdy + wdz is a complete differential. Much later this was distinguished as the case in which a velocity potential existed, or the case of irrotational motion. In a third paper on the motion of fluids l Eider draws attention to a plane irrotational motion of an incompressible fluid, which is characterised by the two conditions du dv __ n dv ___ du dx dy dx By In this connection, Euler acknowledges a debt to d'Alembert for having conceived the device of considering u — iv as a function of x + iy? and u + iv as a function of x — iy.2 (This was before Cauchy had systematised the notion of analytic function, and long before the modern school of hydrodynamics existed.) Eider also writes, with some hint of sarcasm, " However sublime may be the investigations on fluids for which we are indebted to MM. Bernoulli, Clairaut and d'Alembert, they stem so naturally from our two general formulae that one cannot but admire this agreement of their profound meditations with the simplicity of the principles from which I have deduced my two equations, and to which I was directly led by the first axioms of mechanics. " Just because of the analytical difficulties of the general problem, Euler did not misunderstand the importance of the part-experimental, part-theoretical considerations that were used in hydraulics. On the contrary, in that he was personally concerned with the Segner water wheel, had analysed the working of turbines and had himself designed a reaction turbine, he was a pioneer of modern technics. 1 Continuation des recherches sur la theorie du mouvement des fluides, Memoires de V Academic de Berlin, 1755, p. 316. 2 This device is used by d'ALEMBERT in Lis Essai sur la resistance des fluides. HYDRODYNAMICS 305 7. BORDA AND THE LOSSES OF KINETIC ENERGY IN FLUIDS. In this paragraph we shall follow a work of Chevalier de Borda (1733-1799) called Memoire sur Vecoulement des fluides par les orifices des vases.1 Borda's analysis is based on hoth Daniel Bernoulli's hydrodynamics and the mechanics of fluids which d'Alembert had related to his own principle. At first Borda discusses problems of flow, and on each occasion his analysis owes something to Bernoulli and d'Alenxbert. Notable among these problems is the determination of the contracted section — in this connection he considers a re-entrant nozzle, where the contracted section can be calculated and turns out to be equal to half of that of the orifice. But the essential interest of Borda's study is that he drew attention to " hydro dynamical questions in which a loss of living force must be assumed" Such losses appear in a tube which is abruptly enlarged or contracted. With a bold insight, Borda compared the phenomenon which occurs in the fluid to an impact in which a loss of kinetic energy was involved — that is, in the language of the time, to an impact of hard bodies. First Borda establishes the following Lemma, and thus anticipates Carnot's theorem in a special case. " Lemma. — Let there be a hard body a, whose velocity is u, which hits another hard body A whose velocity is V. It is required to find the loss of living force which occurs in the impact. 2 i A r^z 44 Before the impact the sum of the living forces was . After the impact this sum has the value ^ a + A (an + A 2g \ a + A •whence, by difference, aA (u — F)2 -~- Borda considers (see figure) the immersion of a cylindrical vessel into an indefinite fluid OPQR, and seeks the motion which the fluid will have on entering the vessel. He starts from the following con sideration. " The motion of the water in the vessel can be regarded as that of a system of hard bodies that interact in some way. Now we know that the principle of living forces only applies to the motion of such 1 Memoires de VAcademie des Sciences, 1766, p. 579. 20 306 THE PRINCIPLES OF CLASSICAL MECHANICS todies when they act on each other by imperceptible degrees, and that there is necessarily a loss of living force as soon as one of the bodies collides with another. " 0 Fig. 102 In the example with which we are concerned, " the slice mopn which enters the vessel at one instant, occupies the position rsqy at the next instant. It is clear that before it occupies this position the small slice will have lost a part of its motion against the fluid above, as if it had been an isolated mass which had been hit by another isolated mass. But in the case of these two isolated masses there would have been a loss of living force. Therefore there will also be such a loss in the case that we are discussing. " And here is Borda's solution, which follows Daniel Bernoulli's method. Suppose that the fluid has travelled to EF, and that in the next instant it travel to CD. Put AE = x, Ag = a and AB = b. Let u be the velocity of the fluid at E. Assume that the living force of the fluid in the indefinite vessel ROPQ remains zero. Under these conditions, the living force of the fluid in the inner vessel can be written u^bx HYDRODYNAMICS 307 if the living force of the slice which enters the vessel is neglected. " Thus the difference of the living force of all the fluid contained in the vessel will be u2bdx -j- Zbxudu Now while the fluid acquires this increment of living force the slice DCFE, or bdx, is supposed to descend from the height GjB, or a — x. Therefore, if the principle of living forces applied without restriction, it would he true that , . , 7 u2bdx -4- 2bxudu (a — X) bdx = --- " But there is a loss of living force in the whole of the fluid, which arises from the action of the small slice rsqy on the fluid rCDs which is above it. It is easy to see, by the lemma, that if the velocity of the slice opmn is denoted by F, then this loss of living force is badx (V - u)2 _ (F~ u)2 _____ octx • a + dx 2g 2g " Therefore, adding this quantity to the second term of the equation above, the correct solution of the problem is obtained — u*bdx + 2bxudu + bdx (V — u)2 = 2g (a — x) bdx. u It only remains to determine V. For this purpose it is sufficient to observe that the stream of fluid which enters the vessel contracts in the same way as if it left the vessel by the same orifice and entered free space. This must be since, in both cases, the fluid which arrives at the orifice is travelling in the same directions. Now the loss of living force must be distributed from the slice that has the greatest velocity — that is, from that which is at the point of greatest contraction. " Therefore suppose that this point is at o and that m is the ratio of EF to op. Then V = mu, whence u?dx -j- 2xudu + u2dx (m — I)2 = 2g (a — x) dx. This equation is integrated by supposing that x — e and u = o at the beginning of the motion. " Borda then repeats his argument and, this time, follows d'Alembert's method. " What we have just said of the principle of conservation of living forces is also applicable to M. d'Alembert's principle. Not that the 308 THE PRINCIPLES OF CLASSICAL MECHANICS latter principle is always true, for there are some instances in which the way it is applied to the motion of fluids must be somewhat modified. Indeed, we have seen that the slice rsxy only acts on the fluid above in the way that an isolated mass would lose a part of its motion to another mass with which it collided. Whence it follows that in the equation of equilibrium, the accelerating force must not be multiplied by the volume - mt which the slice occupies at the middle to the Zi time interval, but by the volume ot which it occupies at the end of this interval. For the volume ot represents the mass of the small slice and rC represents that of the fluid rCDs. " No purpose would be served by reproducing the calculation which follows, which leads to the same result as the analysis reproduced above. However bold it may have been, Borda's hypothesis is discovered to be in satisfactory agreement with experiment. " A tube 18 lines in diameter and one foot long was made of very uniform tinplate whose edges were tapered. Then, closing the upper orifice with the hand, the tube was plunged into a vessel filled with water. It was assured that the air contained in the tube did not allow the water to enter to the same extent as if both openings had been free. Then the upper orifice of the tube was opened and the water mounted inside the tube to a height greater than its level outside. The experiment was repeated several times and the water rose to its peak which was 4 pouces above the outside level. According to the calculation of M. Bernoulli, it should have risen to 8 pouces. " x The ascent calculated by Borda was 49 % lines. He observed an ascent of 47% lines and attributed the difference to the friction of the fluid on the walls. 1 M£moire$ de I* Academic des Sciences, 1766, p. 147. CHAPTER NINE EXPERIMENTS ON THE RESISTANCE OF FLUIDS (BORDA, BOSSUT, DU BUAT) COULOMB AND THE LAWS OF FRICTION 1. BORDA'S EXPERIMENTS AND NEWTONIAN THEORIES. During the same time that the principles of dynamics were being organised and the foundations of hydrodynamics were being developed, there grew up a complete experimental approach that was determined by requirements of an essentially practical kind. To pause on this remarkable movement is not to move away from the principles of mechanics, for here it can be seen how experiment is dominant in fields where the theory is impotent before the very complexity of even the most tangible phenomena. We shall only deal with some examples of this experimental work in mechanics during the XVIIIth Century. Besides being charac teristic, these examples are ones in which the origins of modern research should be sought, and in which the modest methods deployed (for example, the motive agencies were invariably provided by falling weights) were no obstacle to the application of a rigorous experimental method. But before coming to these examples, it is necessary that we should describe some essays of the theoreticians, who had, indeed, preceded the experimentalists by several years. Newton had developed a schematic theory of fluids, which he considered to be formed of an aggregate of elastic particles which repelled each other, were arranged at equal distances from each other, and were free. If the density of this aggregate was very small, Newton assumed that if a solid moved in the fluid then the parts of the fluid which were driven along by the solid were displaced freely, and did not communicate the motion which they received to neighbouring parts. In this framework, Newton calculated the resistance of a fluid to the translation of a cylinder. He found that this resistance was 310 THE PRINCIPLES OF CLASSICAL MECHANICS equal to the weight of a cylinder of fluid of the same base as the solid, and whose height was twice that from which a heavy body would have to fall in order to acquire the velocity with which the solid moved. The resistance offered to the translation of a sphere, according to the same newtonian theory, is half the resistance which the cylinder encounters under the same conditions. Jean Bernoulli adopted these laws in the discussion of the commun ication of motion which he gave in connection with the controversy on living forces. Newton also formulated a second theory on the resistance of fluids, and applied this to water, oil and mercury. His first theory was only applied to the resistance of air.1 In this second theory, particles of the fluid are contiguous. Newton compares the resistance to the effect of the impact of a stream of fluid on a circular surface, the stream being imagined to leave a cylin drical vessel through a horizontal orifice. He passes to the limit by infinitely increasing the capacity of the vessel, and also the dimensions of the orifice, in order to simulate the conditions of an indefinite fluid. He then substitutes the motion of the circular surface for that of the fluid in the first model of impact. Given this, Newton calculates the resistance offered to the trans lation of a cylinder and finds that the resistance is equal to the weight of a cylinder of fluid whose base is the same as that of the solid and whose height is half that from which a heavy body would have to fall in order to acquire the velocity with which the solid moves in the fluid. This resistance is four times smaller than that provided by the first theory. Further, in the second theory the length of the moving cylinder does not affect the result, for only its base is exposed to the impact of the fluid. Under these circumstances the resistance offered to the translation of a sphere is equal to that which would be offered to the translation of a cylinder circumscribed about the sphere. This result is half that provided by the first theory. The second newtonian theory is applicable to the oblique impact of a stream of fluid on a plane wall. Under these conditions, it leads to a resistance which is proportional to both the square of the velocity and the square of the sine of the angle of incidence. These were the proportions which the experimenters tried to verify. We also add that Daniel Bernoulli, although he did not offer an alternative theory, had already remarked on considerable differences 1 In fact this theory goes back to HUYGHENS (1669), MARIOTTE (1684) and PARTIES (1671). THE RESISTANCE OF FLUIDS 311 between the newtonian laws and experiment.1 Moreover, he legitim ately emphasised that it was necessary to distinguish between the impact of a fluid on a wall and the impact of a fluid on a completely immersed plane.2 Finally, we recall that d'Alembert, in his early work on the resistance of fluids, also calculated the impact of a moveable surface on an infinity of small elastic balls which represented a fluid.3 With this in mind, we come to the experiments of Chevalier de Borda. In the first place, Borda studied the resistance of air.4 By means of a driving weight he made a flywheel rotate and attached plane surfaces of different shapes to the circumference. He took care to correct the results for the friction of the flywheel and to confine the observations to a period of uniform motion, when a steady state had been established. These are Borda's conclusions. 1) The total resistance of the air cannot be calculated as the sum of the partial resistances of each of its elements. For example, the resistance of a circle is not the sum of the resistances of two semicircles. This conclusion is very important — it shoes that the resistance is a phenomenon which behaves integrally, and also makes it clear that the resistance cannot be obtained by an integration which depends on a simple elementary law. 2) The aggregate resistance is proportional to the square of the velocity and the sine of the angle of incidence (not to the square of this sine). Fig. 103 As far as the resistance of water is concerned, Borda confines him self, in this first paper, to the verification of the proportionality to the square of the velocity. 1 Mtmoires de Petersburg, Vol. II, 1727. 2 Ibid., Vol. VIII, 1741. 3 See above, p. 295. 4 Memoires de V Academic des Sciences, 1763, p. 358. 312 THE PRINCIPLES OF CLASSICAL MECHANICS Borda returned to tlxe resistance of fluids in a paper dated 1767.1 He worked with a circular vessel 12 feet in diameter. By means of driving weights varying from 4 ounces to 8 pounds, he made a sphere of 59 lines diameter move through the water. The sphere was made of two equal parts, which could be joined together or separ ated as desired. When working with one hemisphere, Borda allowed it to present either the section of a great circle or the convex part, to the fluid. Borda took care to allow for " the friction and the impact of the air on the flywheel " by making the apparatus rotate freely without the sphere. He verified that the resistance was very accurately proportional to the square of the velocity. In addition, he established that the resistance of the hemisphere was nearly independent of the surface that was presented to the fluid. From this he concluded that " at these small velocities., the forward part of the body is the only one ichich has resistance. " Borda next turned his attention to the absolute magnitude of the resistance, and compared the values observed with those calculated from what we have called the second newtonian theory (Principia, Book II, Proposition XXXVIII). He found that the resistance of the hemisphere when it offered a section of a great circle to the fluid was 2% times as great as the resistance of the whole sphere, itself accur ately equal to the resistance of the hemisphere when this offered its convex side to the fluid. Now, according to the newtonian theory, the first resistance is twice the second. The disagreement is evident. Similarly, Borda determined the oblique resistance. He established, exactly as in his experiments in air, that the law of the square of the sine was not true, and even declared "that when the angles of incidence are small the resistance does not decrease as much as the simple sine. " Borda also studied the influence of the depth on the resistance in water. He established that the resistance decreased with the depth, and that, at the surface, it increased more rapidly than the square of the velocity. In this connection, he attempted an explan ation which was only half convincing, by falling back on his own theory of the losses of living force in fluids.2 " It is clear that when the sphere is only 6 pouces below the surface it does not impart such great velocities to the neighbouring parts as when it moves in the surface of the water. For in the first case the fluid is free to run round the whole circumference of the sphere 1 Memoires de VAcademie des Sciences, 1767, p. 495 2 See above, p. 305. THE RESISTANCE OF FLUIDS 313 while in the second, it cannot escape along the upper part of the sphere. From which it follows that in the first instance the fluid neither gains nor loses as great a quantity of living forces as in the second. " Borda then worked with a model ABCD in which AH = HD = 6 pouces and J3C = 4 pouces. The difference between the resistance when the side A (angle BAC), and then the side D (two arcs of circles, BD and DC, with centres on J3C), were offered to Pig. 104 the fluid was negligible — the newtonian theory predicted a ratio of 28 to 15 for these resistances. Borda's general conclusion was that the newtonian theory could not account for the resistances of fluids. " The ordinary theory of the impact of fluids only gives relationships which are absolutely false and, consequently, it would be useless and even dangerous to wish to apply this theory to the craft of the construction of ships. " 2. THE ABBE BOSSUT'S EXPERIMENTS. In 1775 Turgot asked the Academie des Sciences " to examine means of improving navigation in the Realm. " A committee consisting of d'Alembert, Condorcet and the Abbe Bossut (as secretary) immediat ely took up the investigation and, between July and September, 1775, conducted numerous experiments " on a large stretch of water in the grounds of the Military College. " They secured the cooperation of the mathematicians attached to that College, including Legendre and Monge. The committee reported to the Academie des Sciences on April 17th, 1776, and this report, Nouvelles experiences sur la resistance des fluides was published at Paris (Jombert) in 1777, under the names of the three members of the committee. The experimental method is referred to in the following terms. " To ask questions of nature by doing experiments is a very delicate matter. In vain do you assemble the facts if these have no relation to each other ; if they appear in an equivocal form ; if, when they are produced by different causes, you are unable to assign and distin guish the particular effects of these causes with a certain precision. . . . Do not heed the limited experimenter, the one who lacks principles ; guided by an unreasoning method, he often shows us the same fact in different guises — of necessity, and perhaps without recognising this himself; or he gathers at random several facts whose differences 314 THE PRINCIPLES OF CLASSICAL MECHANICS he is unable to explain. A science without reasoning does not exist or, what comes to the same thing, a science without theory does not exist. " * Bossut explicitly distinguished between the resistance of fluids that were indefinitely extended (a ship on the sea or on wide and deep rivers) and the resistance in narrow channels (shallow or narrow rivers and canals). Borda's experiments were conducted in fluids that were, for practical purposes, indefinitely extended. On the other hand, in order to study the effect of the depth of immersion, Franklin had worked on a small scale with a canal and a model of a ship which was 6 pouces long, and 2% pouces wide.2 The basin at the Military College was 100 feet long and 53 feet wide at the centre, its maximum depth being 6% feet. A weight hung over a pulley assured the traction of the model, which was equiped with a rudder in order that its motion might be determinate. Bossut's Model No. 1. Fig. 105 Bossut used twelve different models of ships and carried out a total of about 300 trials, of which about 200 were in an effectively indefinite fluid and the remainder in an artificially constructed channel whose depth and width were variable at will. When he compared the experimental results with the second new- tonian theory, Bossut came to the following conclusions. 1) On a given surface, and at different velocities, the resistance is " approximately in the square ratio, just as much for oblique impacts as for direct impacts. More accurately, the resistance increases in a greater ratio than the square. " He gives the following explanation of this fact. " The fluid has greater difficulty in deflecting itself when the velocity increases — it piles up in front of the prow and is lowered near the stern. " 3 2) " For surfaces which are equally immersed in the fluid and only different in respect of their width, the resistance sensibly follows 1 Nouvelles experiences sur la resistance des fluides, p. 5. 2 CEuvres completes de Franklin, Vol. II, p. 237. 3 BOSSUT, op. cu., p. 147. THE RESISTANCE OF FLUIDS 315 the ratio of the surfaces. . . . More precisely, it increases in a ratio which is a little greater than that of the extent of the surfaces. " 1 " The resistance of bodies which are entirely submerged is a little less than that of bodies which are only partly submerged. " 2 3) " The law of the square of the sine is less justified when the angles are very small. " 3 In order to express the results of these experiments, the Abbe Bossut chose a provisional law of the form sin"£ where i is the angle of incidence. He found that the exponent n varied from 0.66 to 1.79, according to the model studied. This led him to conclude — " The resistances which occur in oblique impacts cannot be explained by the theory of resistances by introducing, instead of the square, some other power of the sine of the angle of incidence in the expression for the resistance. " 4 In order to determine the magnitude of the resistance of water, Bossut made two corrections. The first depended on the friction of the pulley which supported the cable and the motive weight — he measured this friction by varying the motive weight. The second correction arose because of " the impact of the air " on the model. Indeed, to the author, resistance was an impact phenomenon. He was guided throughout by the second newtonian theory. In order to eli minate the impact of the air, Bossut measured the surface of the model which was offered to the impact of air, and assumed that the impacts of the water and the air on the model were respectively "in compound proportion to the impacted surfaces and the densities of the two fluids." Having made these two corrections, Bossut concluded — " The resistance perpendicular to a plane surface in an indefinite fluid is equal to the weight of a column of fluid having the impacted surface as its base and whose height is that which corresponds to the velocity with which the percussion occurs. " 5 Bossut tried to analyse further the phenomenon of resistance ; he sought to emphasize the part played by the " tenacity " of the fluid and the " friction caused along the length of the boat by the water. " From this somewhat arbitrary decomposition, he felt justified in drawing the following conclusions : " We have observed that as soon as the friction is overcome, the 1 BOSSUT, op. cit., p. 152. 2 Ibid., p. 157. 8 Ibid., p. 163. 4 Ibid., p. 164. 5 Ibid., p. 173. 316 THE PRINCIPLES OF CLASSICAL MECHANICS slightest force sets the hoat in motion. From which we have concluded that the tenacity of the water is extremely small and that this resistance must be considered absolutely nil in comparison with that caused by inertia. The same applies to the friction of the water along the sides and bottom of the boat. This friction is very slight and its effect cannot be distinguished from that of the pulleys or of the resistance of the air. " x Again Bossut noted the resistance in a narrow canal, superior to the resistance in an unlimited fluid, and he underlined the influence of the transversal dimensions and of the form of the vessel used for comparison. For the construction itself of the canals, his paper is limited to cautious generalities : the canal should be as large and as deep as possible, " without nevertheless going to superfluous expense " ; subterranean canals should be avoided unless local circumstances make their use indispensable. Indeed, concludes this sagacious rapporteur, " a canal is an object of utility and not an instrument for ostentation. " 3. Du BUAT (1734-1809). HYDRAULICS AND THE RESISTANCE OF FLUIDS. Du Buat began by directing the construction of fortifications and on this occasion was the promotor of * geometric cotee '. He later devoted himself to hydraulics, as " Captain of the infantry, engineer to the King." The Principes tfhydraulique, the first edition of which is dated 1779, deals with " the motion of water in rivers, canals and conduits ; the origin of rivers and the formation of their beds ; the effect of locks, bridges and reservoirs 2 ; of the impact of water ; and of navigation on rivers as well as on narrow canals. " Du Buat wrote, " there is no argument which can be used to apply the formulae for flow through orifices to the uniform flow of a river, which can only owe the velocity with which it moves to the slope of its bed, taken at the surface of the current. " Gravity is, on both cases, certainly the cause of the motion. " I therefore set out to consider whether, if water was perfectly fluid and ran in the part of a bed which provided no resistance, it would accelerate its motion like bodies which slide on an inclined plane. . . . Since it is not so, there exists some obstacle which prevents the accelerating force from imparting fresh degrees of velocity to it. Now, of what can this obstacle consist, except the friction of the water against the walls of the bed and the viscosity of the fluid ? " And Du Buat stated this principle — " When water runs uniformly in some bed, the force which is necessary to make it run is equal to 1 BOSSUT, op. cit., p. 173. 2 Read " weirs. " THE RESISTANCE OF FLUIDS 317 the sum of the resistances to which it is subject, whether they are due to its own viscosity or to the friction of the hed. " In the 1786 edition of his Traite, Du Buat amends this statement and no longer speaks of the viscosity, but only of the resistance of the bed or the containing walls. The viscosity only enters indirectly, 44 in order to communicate the retardation due to the walls, step by step, to those parts of the fluid which are not in contact with them. " This effect only influences " the relation between the mean velocity and that which is possessed by the fluid against the walls. " In canals of circular or rectangular section, Du Buat introduced the notion of mean radius (the ratio of the area of the cross-section to the length of the perimeter in contact with the fluid) and evaluated the resistance of the walls to unit length of the current by the product of this radius and the friction on unit surface. He assumed that the resistance of the bottom was proportional to the square of the velocity of the current, and likened it to the impact of the water on the irre gularities on the bottom. Du Buat did not confine himself to this theoretical outline but, like the Abbe Bossut, he sought experimental confirmation. The second edition of his Traite is concerned with these experiments. Du Buat verified that the friction of fluids was independent of their pressure. This he did by making water oscillate in two siphons of very different depth. He investigated the friction of fluids on different materials (glass, lead and tin) and, having observed that this friction was always the same, he assumed that the water " itself prepares the surface on wliich it runs " by wetting the pores and cavities as a varnish does. He went further, and even held that the resistance of the walls did not depend on their roughness — a conclusion that was very far from being correct. In order to obtain the resistance of the walls, Du Buat worked with an artificial canal of oak planks, whose section could be varied in shape and size. He also worked with pipes of tinplate or glass of very different diameters. He discovered that the resistance of the walls was in a smaller ratio than the square of the velocity x and 1 DE PHONY advocated a formula for the resistance which had the form av -f bv2. Much later, after having observed the oscillations of a circular plate in a fluid medium, COULOMB was to say, " There must be two kinds of resistance. One, due to the coher ence of the molecules which are separated from each other in a given time, is propor tional to the number of these molecules and, consequently, to the velocity. The other, due to the inertia of the molecules which are stopped by the roughnesses with which they collide, is proportional to both their number and their velocity and, consequently, to the square of their velocity. " COULOMB was, before STOKES, the first to hold that the velocity of a viscous fluid relative to a solid was nothing at the surface of contact, and that it then varied continuously in the fluid. 318 THE PRINCIPLES OF CLASSICAL MECHANICS gave an empirical formula for this resistance which was only surpassed in accuracy by those of Darcy (1857) and Bazin (1869). Du Buat then turned to the empirical relationships between the mean velocity, the velocity at the centre of the surface and the velocity at the centre of the bed. To account for the resistance due to bends, Du Buat assumed a series of impacts on the banks, and expressed the resistance as a number proportional to the square of the mean velocity, the square of the sine of the angle of incidence and the number of " ricochets. " He applied his empirical formula to the eddies and local variations of level which are found upstream from barrages and narrows by considering small consecutive lengths of the current. Du Buat also treated the decrease of the slope, and the increase of the depth, from the source of a river to its mouth. He took account of tributaries, temporary and periodic floods, changes of course, the retarding effect of the wind and even the influence of the weeds which grew in the bottom. In order to ascertain the resistance of fluids to the translation of a solid,, Du Buat exposed a tinplate box to the current. The box was either cylindrical or in the form of a parallelipiped whose edges were parallel to the flow lines. The boxes were provided with holes which could be opened or closed at will. A float allowed the difference of the levels, outside and inside the box, to be measured, and thus the pressures at different points of the surface of the box to be estimated. In this way Du Buat showed the existence of an over-pressure at the front (with respect to the previously existing state, in which the level was uniform) and a " non-pressure, " or suction " at the back acting in the same direction as the over-pressure. " The total observable resistance corresponds to the sum of these two effects. Du Buat measured them separately, and showed that the over-pressure was approximately the same for a thin plate, for a cube and for a parallelipiped. On the other hand, the " non-pressure " decreased rapidly when the solid became relatively longer. Du Buat found that the resisting force of a fluid mass to a solid in translation was less than the resistance of the solid at rest to the moving fluid, if the relative velocity was the same in both cases. This is ex plained by the fact that he worked on a limited fluid mass. Du Buat then set out to measure the amount of the fluid which accompanied a solid in its motion through a practically indefinite fluid. He made a solid oscillate, like a pendulum in the fluid, and studied the variation of the amplitude of small oscillations — a consequence of the decrease of the weight of the solid body due to the upthrust of the fluid, and the increase of its mass due to the mass a fluid carried along. If p THE RESISTANCE OF FLUIDS 319 is the weight of the oscillating body (weighed in the fluid), P the weight of fluid displaced, nP the sum of the weights of the fluid displaced and the fluid carried along, / the length of the pendulum and a the length of an isochronous pendulum in the vacuum, then I T_ P I a 1\ whence n = 4r ( — II- a p+nP " P U Indeed, in the fluid P + nP pg *L J -, —_ n ^T «, = £-2_ — y = ,-~n g r p+nP and also Du Buat estabKshed, by working in water with metallic bodies and in air with distended balloons, that the amount of fluid carried along by solids was approximately proportional to the resistance obtained by other methods. Further, he suggested extending the measurement of oscillations in order to determine the resistance of fluids, by working with pendulums consisting of long columns, so that the curvature of the trajectory might be a minimum. From all these investigations, which place Du Buat among the greatest experimenters of his time, the author concludes that he has " not done much more than destroy the old theoretical structure, " and he appealed for more experiments, in the hope that a more correct theory might emerge from them. 4. COULOMB'S WORK ON FRICTION. Coulomb was not the first to make experiments on the friction of sliding and the stiffness of ropes. Amontons, in 1699 1, had stated that the friction was proportional to the mutual pressure of the parts in contact. Muschenbroek intro duced the amount of the area of contact. De Camus, in a Traite des forces mouvantes, and D^saguillers in a Cours de physique, remarked that the friction at rest was much greater than the friction in motion. In connection with the stiffness of ropes, Amontons showed that the force necessary to bend a rope round a cylinder was inversely proportional to the radius of the cylinder and directly proportional to the tension and the diameter of the rope. 1 Memoires de PAcademie des Sciences, 1699. 320 THE PRINCIPLES OF CLASSICAL MECHANICS In 1781 the Academie des Sciences chose the subject of the laws of friction and the stiffness of ropes for a competition, asking for a return " to new experiments, made on a large scale and applicable to machines valuable to the Navy, such as the pulley, the capstan and the inclined plane. " Coulomb, who was then senior captain of the Royal Corps of En gineers, won the prize with his Theorie des machines simples en ayant egard au frottement et a la roideur des cordages.1 In the frontispiece of his paper, Coulomb quotes this saying of Montaigne — " Reason has so many forms that we do not know which to choose — Experiment has no fewer " (Essais, Book III, Chapter XIII). In fact Coulomb's work is a model of experimental analysis, carried out with precision and exemplary detail, and from which he obtained a theory applicable to machines. The parameters which Coulomb used in his study of friction were the following — the nature of the surfaces in contact and of their coatings ; the pressure to which the surfaces are subject ; the extent of the surfaces ; the time that has passed since the surfaces were placed in contact ; the greater or lesser velocity of the planes in contact ; and, incidentally, the humid or dry condition of the atmosphere. He described his apparatus in great detail and, for example, mention ed " a plank of oak, finished with a trying-plane and polished with seal-skin. " He studied the friction of oak on oak, " seasoned, along the grain of the wood, with as high a degree of polish as skill could achieve. " All the result obtained were recorded, experiment by experiment, with the rigor of an official report. He first studied the friction of sliding between two pieces of seasoned wood (oak on oak, oak on fir, fir on fir, elm on elm). He then studied the friction between wood and metals, between metals with or without coatings, etc. . . . By way of an example, here is a summary of some of his conclusions. " 1. The friction of wood sliding, in the dry state, on wood opposes a resistance proportional to the pressures after a sufficient period of rest; in the first moments of rest this resistance increases appreciably, but after some minutes it usually reaches its maximum and its limit. " 2. When wood slides, in the dry state, on wood, with any velocity, the friction is once more proportional to the pressures but its intensity is much less that which is discovered on detaching the surfaces after some moments of rest. " 3. The friction of metals sliding on metals, without coatings, 1 Memoires des Savants etrangers, Vol. X. THE RESISTANCE OF FLUIDS 321 is similarly proportional to the pressures but its intensity is the same whether the surfaces are detached after some moments of rest, or whether they are forced into some uniform velocity. " 4. Heterogeneous surfaces, such as wood or metals, sliding upon each other without coatings, provide, in their friction, very different results from the preceding ones. For the intensity of their friction, relatively to a time of rest, increases slowly and only reaches its limit after four or five days, or even more. , . . Here the friction increases very appreciably as the velocities are increased, so that the friction increases approximately in an arithmetic progression when the velo cities increase according to a geometric progression. " The most debatable part of Coulomb's paper is that in which he attemps to construct a model of the production of friction. 46 The friction can only arise from the engaging of the projections from the two surfaces, and coherence can only affect it a little. . . . The fibres of wood engage in each other as the hairs of two brushes do ; they bend until they are touching without, however, disengaging ; in this position the fibres which are touching each other cannot bed themselves down further, and the angle of their inclination, depending on the thickness of the fibres, will be the same under all degrees of pressure. Therefore a force proportional to the pressure will be necess ary for the fibres to be able to disengage. " At first Coulomb used the same arrangement as Amontons for the investigation of the stiffness of ropes. Later he developed a new one which allowed him to work with more industrial cables, namely, " ropes of three untarred strands. " He summarised the effect of the stiffness of ropes by means of the formula A + BT R where A = hrq, B = h'r** where R is the radius of the pulley, r the radius and T the tension of the rope. The exponents q and ^ are approximately equal. The mechanics of friction was still a very skeletal one in Coulomb's paper. Coulomb assumes that, in order to draw a weight P along a horizontal plane, it is necessary to deploy a force T= A + — P In this formula, A is a small constant depending on the " coherence " of the surfaces and JJL is a coefficient (the reciprocal of the coefficient 322 THE PRINCIPLES OF CLASSICAL MECHANICS of friction which is now commonly used) depending on the nature of the surfaces. Turning his attention to the observations made of the launching of ships at the port of Rochefort in 1779, Coulomb calculated the Fig. 106 force necessary to hold a body on an inclined plane. He obtained the result that __ AJLL + P (cos 7i + ju, sin n) H cos m + sin m where n is the inclination of the plane and m the angle between the force T and the plane BC. From this he easily deduced that T is a minimum « cos m lor u = — . sin m The mechanics of friction was born of some experiments in physics in the XVIIth Century and then, for an essentially practical purpose, was systematised by Coulomb. But, at the time, it remained linked to the common practice of engineering, while rational mechanics developed, without regard to friction, in the mathematical field. CHAPTER TEN LAZARE CARNOTS MECHANICS 1 . CARNOT AND THE EXPERIMENTAL CHARACTER OF MECHANICS. In 1783 Lazare Carnot (1753-1823) published an Essai sur les machines en general. He later extended this under the title of Principes generaux de Fequilibre et du mouvement (1803). In this interval La- grange published the first edition of his Mecanique analytique (1788). But Carnot's ideas varied so little from, the Essai to the Principes that it can be maintained that Lagrange had no influence on Carnot. Further, it is natural to think of Carnot as a predecessor of Lagrange, in spite of details of simple chronology.1 In the field of principles, we are indebted to Carnot because he was the first to assert the experimental character of mechanics — universally accepted now. This is cpiite in contrast with the ideas professed by Euler, and more often, by d'Alembert. The declarations which follow are taken from the Principes and are to be contrasted, in particular, with the introduction to d'Alembert's treatise. " The Ancients established the axiom that all our ideas come from our senses ; and this great truth is, today, no longer a subject of controversy. . . . [Here Carnot is invoking Locke's Essay on Human Understanding.] " However, all the sciences do not draw on the same experimental foundation. Pure mathematics requires less than all the others ; next come the physico -mathematical sciences ; then the physical sciences. . . . " Certainly it would be satisfactory to be able to indicate exactly 1 CARNOT himself wrote, in the preface to the Principes, " Since the first edition of this work in 1783, under the title of Essai sur les machines, there have appeared, in all branches of mechanics, works of such beauty and of such scope that there hardly remains room for some remembrance of mine. However, as it contained some ideas that were new at the time it appeared, and as it is always valuable to contemplate the fundamental truths of science from the various points of view that can be chosen, a new edition has been asked of me. ..." 324 THE PRINCIPLES OF CLASSICAL MECHANICS the point at which, each science ceased to be experimental and became entirely rational [read, in order to develop rationally, starting from principles obtained from experiment] ; that is, to be able to reduce to the smallest number the truths that it is necessary to infer from experiment and which, once established, suffice to embrace all the ramifications of the science, being combined by reason alone. But this seems to be very difficult. In the desire to penetrate more deeply by reason alone, it is tempting to give obscure definitions, vague and inaccurate demonstrations. It is less inconvenient to take more in formation from experiment than would strictly be necessary. The development may seem less elegant. But it will be more complete and more secure. . . . " It is therefore from observation that men derived the first concepts of mechanics. However, the fundamental laws of equilibrium and motion which serve as its foundation offer themselves so naturally to reason on the one hand, and on the other, show themselves so clearly in the most common facts, that it is difficult to say whether it is from the one rather than from the other that we derive our perfect conviction of these laws ; and whether this conviction would exist without the con currence of these laws with the first. These facts seem too familiar for us to be able to know at what point, without them, reason alone would be able to establish definitions. And, on the other hand, if reason is unable to connect these facts by analogy, they appear too isolated for us to be able to weld them into principles. " l 2. THE CONCEPTS AND POSTULATES OF CARNOT'S MECHANICS. Carnot had certainly studied Euler and d'Alembert, and thus knew of the theory of forces and also of that of motions (in the purely kinematic sense). He reproaches the first for " being founded on a metaphysical and obscure notion of forces. " If, on the contrary, the word force is understood to be the momentum impressed on a system, the first theory reduces to the second and requires an appeal to experiment. At least in principle, Carnot adopts the second attitude and seeks to reduce mechanics to the study of the communication of motion. He applies the laws of mechanics to the reasoned observation of problems of impact. He then reduces the action of a continuous force to that of a series of infinitely small impacts. " Weight and all forces of the same kind act in imperceptible degrees and produce no sudden changes. However, it seems rather 1 Principes generaux de Vequilibre et du Tnouvement, p. 2. LAZAKE CARNOT 325 natural to consider them as dealing infinitely small blows, at infinitely short intervals, to the bodies which they actuate. " Thus the fundamental law of Carnot's mechanics is written, apart from notation, in the form Fdt = d(mv). But Carnot accompanied this fundamental law with the following commentary. u At first I shall repeat that the question here is not that of the original causes which create motion in bodies, but only that of the motion already produced and inherent in each of them. The quantity of motion already produced in a body is called its force or its power. Thus the forces which are considered in mechanics are not metaphysical or abstract entities. Each of them resides in a determinate mass. The force is the product of this mass and the velocity which the body takes if it is not obstructed by the motions of other bodies which are incompatible with its own. Such incompatibility makes some bodies lose a part of their quantity of motion ; it makes others add to it, and creates it in those which had none. Each body assumes a kind of combined velocity, in between the one which it must have already had and those which are newly impressed on all its parts. Now it is this compound velocity that it is necessary to determine, at each instant and for each point of the system, when the shapes of the different parts which compose it, their masses and the velocities which they are supposed to have received previously — whether by earlier impacts or by external agencies of any kind — are known. Thus, in a word, we do not seek the laws of motion in general, but rather the laws of the communication of motion between the different material parts of a single system. " l In fact, Carnot did not rigorously dispense with the concept of force. It may even be said that he multiplied the names for it, as we shall see. Moreover, this conforms with his general attitude — his mechanics did not depend on a closed set of axioms. Carnot variously called the product of a body's mass and the accelerating force [read " acceleration"] its motive force, force of pressure or dead force. Thus gravity or heaviness is an accelerating force and weight, a motive force. By moving force Carnot understood " the motive force applied to a machine in order to overcome the resistances, or to produce any motion at all. " If the living force is expressed by the product Trav2, the latent living force is expressed by the product PH of a weight and 1 Principes gGneraux de Fequilibre et du mouvement, p. 47. 326 THE PRINCIPLES OF CLASSICAL MECHANICS a height. The elementary work of a force is called, hy Camot, the moment of activity achieved by a motive force. As for the moment of absolute activity of a moving body, this can be expressed in modern language by the product mv (v + dv) where v + dv is the velocity of the body at the time t + dt (if the motion is continuous) . In impact, the same moment of activity would be written TTii; (v -f- Av) where Av is a finite increment. Carnot next introduces the force of inertia by means of the following definition — " The resistance offered by a body to a change of state " or the " reactions opposed to a system of bodies which make it pass from rest to motion. " For example, in an impact (the external actions being supposed negligable) the force of inertia of a body of mass m whose velocity changes from tT0 to t^ would be, in Carnot's sense, m (£TQ — JTJ. Here the force of inertia coincides with the quantity of motion lost. But, in general, the quantity of motion lost is the "re sultant of the quantity of motion produced by the motive force and the quantity of motion produced by the force of inertia. " Finally, Carnot understands the force exerted on a body of the system to be the resultant of the motive force and the force of inertia. In passing, we note a curious discussion on this subject. In his Sixty-Sixth Letter to a German Princess, Euler had criticised the ex pression ** force of inertia " as uniting the concept of force (capable of changing the state of a body) and the word inertia (express ing the property of a body that tends to preserve it in its state). Carnot objected that " the inertia is merely a property which may not be introduced in the calculations, while the force of inertia is a real measurable property ; it is the quantity of motion, which this body imparts to any other body, that displaces it from its state. " * Carnot assumed the following postulates as a foundation for his mechanics. 1) The principle of inertia. 2) A system in equilibrium remains in equilibrium under the application of forces which are in equilibrium among themselves. 3) In a system of forces in equilibrium, each force is equal and opposed to the geometric sum of all the others. 4) " The quantities of motion of motive forces which, in a system of bodies, destroy each other at all times, can always be decomposed 1 Principes g$neraux de requilibre et du mouvement, p. 73. LAZARE CARNOT 327 into other forces which are, taken in pairs, equal and directly opposed along the direction of the straight line which connects the two bodies to which they belong. And, in each of these bodies, each force can be regarded as nullified by the action of the other. " 5) The action of one body on another by impact, traction or pressure, only depends on the relative velocity of the bodies. 6) " The quantities of motion or the dead forces which the bodies impress on each other through threads or rods are directed along these threads or rods ; and those which they impress on each other by impact or pressure are directed along the perpendicular erected at their common surface at the point of contact. " 7) Hypotheses expressing the laws of inelastic, elastic and partially- elastic impact. Given these definitions, Carnot introduced the concept of geo metrical motion into mechanics in the following way. " Every motion which is imparted to a system of bodies and which does not alter the intensity of the action which they exert or could exert on each other when any other motions whatever are imparted to them, will be called a geometrical motion. Then the velocity which each body assumes will be called its geometrical velocity. " l Carnot has the following comment to make about this concept. " This denomination of geometrical motion is based on the fact that the motions concerned have no effect on the action which can be exerted between the bodies of the system, and that they are inde pendent of the rules of dynamics. . . , They only depend on the con ditions of constraint between the parts of the system and, consequently, can be determined by geometry alone. 46 The theory of geometrical motions is, in a sense, a science inter mediate between geometry and mechanics. It is the theory of the motions that a system of bodies can assume without the bodies hinder ing each other, or exerting any action or reaction on each other. " 2 In modern language, Carnot's geometrical motions are virtual dis placements (finite or infinitely small) compatible with the constraints between the bodies of the system. 3. CARNOT'S THEOREM. In the second part of his Principes fondamentaux Carnot studied the motion of systems, taking as his basis the problems of impact between " hard bodies " — that is, bodies devoid of elasticity. 1 Principes generaux de requilibre et du mouvement, p. 108. 2 Ibid., p. 106. 328 THE PRINCIPLES OF CLASSICAL MECHANICS Carnot first shows that " if a system of hard bodies suffers an impact or any instantaneous action, either directly or by means of some mechanism without elasticity, the motion taken by the system is necessarily a geometrical one. " Indeed, if the bodies contiguous with the system by which the action is propagated are considered in pairs, after the impact they have no relative velocity in the line of their reciprocal action. Their real motions after the impact cannot therefore produce any action between them. It follows that the motion of the system after the impact is necessarily a geometrical one. Moreover, it is easy to see that every geometrical motion which is imparted to any system is received by the system without alteration. Turning to the consideration of a system of hard bodies which sustains an impact, Carnot decomposes (after the manner of d'Alembert) the motion of the system before the impact into two others. The first of these is that which remains after the impact and the second is, consequently, necessarily destroyed by the impact. If only the first motion is imparted to the system, it will necess arily be received without alteration. Under the influence of the second motion, also considered in isolation, the system remains in equilibrium. Carnot writes, " This is what constitutes d'Alembert's famous principle. But it must be recalled that it is only applicable to perfectly hard bodies and to mechanisms without elasticity — this, I think, has not been observed explicitly before. If the bodies were elastic, the motion before the impact would decompose into two in the same way as for hard bodies. One of these motions would be the motion that remains after the impact and the other would be destroyed. But the independence of these motions would not subsist ; for if the first alone were suppressed, there would not be equilibrium. This inde pendence of the two motions is based on the fact that the motion after the impact is geometrical ; that is, it does not tend to increase or decrease the intensity of the impact, and it is only such because the bodies, being hard, etc. ..." Let U denote the velocity lost by a particle M during the impact and let V be its velocity after the impact. By induction, starting from TorricellVs principle, Carnot states the law (1) $MUVcos(vTU) = Q. Here indeed, Carnot makes appeal to continuous motions by starting from the axiom that " when the centre of gravity is lowest, the system IAZARE CARNOT 329 is in equilibrium. " If p is the accelerating force, Carnot writes the condition for the equilibrium of a system, in continuous motion under the influence of the forces p, in the form SpMFcos (pTV) = 0. From this he deduces the law (1) by applying this principle to percussions. Carnot verifies the law (1) for the particular impact of two hard bodies, using an analysis that is, this time, direct. He then extends the law to the impact of any number of hard bodies. From these results, Carnot easily deduced the following theorem, with which his name is still associated. " In the impact of hard bodies, the sum of the living forces before the impact is always equal to the sum of the living forces after the impact together with the sum of the living forces that each of these bodies would have if it moved freely with only the velocity which it lost in the impact. " * Indeed, it was sufficient for him to write SMTF2 = SMF2 + SM U2 + 2 $MVU cos (VTU] where W is the velocity before the impact and law (1) is applied. Using d'Alembert's procedures throughout, Carnot treated problems of elastic impact as corollaries of problems of impact between " hard " bodies. The elasticity doubles the momentum lost without changing its direction. Thus, to Carnot, the conservation of living forces in the impact of perfectly elastic bodies is justified by his theorem on the impact of hard bodies. From the general equation (1) Carnot also deduced the remarkable result that the sum of the living forces due to the velocities lost is a minimum in the impact of a system of hard bodies. " Among the motions to which a system of perfectly hard bodies is susceptible, when the bodies act on each other by a direct impact or by any mechanism without elasticity, so that there results a sudden change in the state of the system, the one that actually remains after the action is the geometrical motion which is such that the sum of the products of each of the masses by the square of the velocity that it loses is a minimum ; that is, less than the sum of the products of the masses and the square of the velocity that it would have lost if the system had acquired any other geometrical motion. " Carnot himself remarked that this result was directly connected with Maupertuis9 application of the principle of least action to the impact of bodies. 1 Principes genfraux de Fequilibre et du mouvement^ p. 145. 330 THE PRINCIPLES OF CLASSICAL MECHANICS In this connection, Carnot emerges as an opponent of the doctrine of final causes. Indeed, he declares that his demonstration of this minimum law " is more general [than that of Maupertuis] because it includes bodies which have various degrees of elasticity. But it also demonstrates how insecure are those which are based on final causes, since it shows that the principle is not general, but restricted to systems of bodies which have the same degree of elasticity. " Without carrying this analysis of Carnot's mechanics further, we shall indicate how he passed from the study of these problems of impact to problems in which continuous forces intervene. " When a system of hard bodies, free or acted upon by any mechanism without elasticity, and actuated by any moving forces, changes its motion by imperceptible degrees then if, it any instant of the motion, each one of the particles is catted m ; its velocity V ; its motive force P x ; the velocity that it would take if the actual motion were suddenly suppressed and replaced by another geometrical one, u ; the element of time, dt ; then there will obtain Smud [V cos (u^V)] — SmuPdt cos (u^P] = 0. " This theorem is deduced from the general formula (1) by observing that Pdt cos (£TP) — d[V cos (uTP)] is the projection, on the direction of u, of the velocity lost by the mass m, due to the action of the other elements of the system. Carnot also develops some very interesting considerations on the work of the internal forces in animal systems. " An animal, like the inanimate bodies, is subject to the law of inertia. That is, the general system of parts which compose it cannot produce by itself any progressive motion in any direction. ... In the whole system of the animal, the principle of the equality of the action and the reaction is applicable, as in inert matter. So that it is only by the friction of its feet on the ground that it can carry itself forward, thereby impressing on the earth on which it walks a quantity of motion equal and opposite to that which it assumes, but which is imperceptible to us. " It therefore seems, as far as its physique is concerned, that the animal may be considered as an assembly of particles separated by springs which are more or less compressed and which, by this fact, 1 Here it is necessary to read " accelerating force. " LAZARE CARNOT 331 store a certain quantity of living forces ; and that these springs, by extending, may be considered to convert this latent living force into real living force. . . . " When a similar agency imparts living force to its own mass, al though the quantity of motion which results in any direction may be zero, the living force is not zero. And if this agency is applied to a machine, its acquired living force will be, by means of this machine, transmitted to the resisting forces without loss — always with the reservation that there should be no impacts ; for what will be consumed will be wholly absorbed and will be precisely what we call the effect produced. " 1 The general conclusion of Carnot's mechanics is the following one. " For any system of bodies, animated by any motive forces, in which several external agents such as men or animals — either by themselves or by machines — are used to move the system in different ways, whatever may be the change produced in the system, the moment of activity consumed by the external powers in any time will always be equal to half the amount by which the sum of the living forces in the system of bodies to which they are applied will be increased during this time, less half the amount by which this same sum of living forces would be increased if each of the bodies had moved freely on the curve which it described — supposing that it had experienced the same motive force, at each point of this curve, as that which it actually experienced ; and provided always that the motion changes by imper ceptible degrees, so that, if machines with springs are used, these springs are left in the same state of tension as at the beginning. " Certainly Carnot's language did not approach the clarity of the great authors of the Century, But the foundation of his work is of an undisputed originality, at once physical and philosophical. In fact, Lazare Carnot was to inspire Laplace, Barre de Saint- Venant and probably Coriolis as well. 1 Principes ggneraux de Vequilibre et du mouvement, p. 246. CHAPTER ELEVEN THE " MECANIQUE ANALYTIQUE " OF LAGRANGE 1. THE CONTENT AND PURPOSE OF LAGRANGE'S " MECANIQUE ANALYTIQUE. " We now come to a piece of work which, united and crowned all the efforts which were made in the XVIIIth Century to develop a rationally organised mechanics. Coming from a Touraine family, Louis de Lagrange (1736-1813) started his career at Turin, where he had been born. After having come under the influence of Euler at the Academy of Berlin, he finally went to Paris in 1787 where, in particular, he inaugurated the teaching of analysis at the £cole polytechnique. Thus, by his descent and for an important part of his scientific career, Lagrange belonged to France. The first edition of the Mecanique analytique appeared in 1788,1 In it Lagrange accomplished the project, which had been conceived and partially executed by Euler, of a single treatise of rational science (analytice exposita) covering all branches of mechanics, statics and hydrostatics, dynamics and hydrodynamics. Lagrange's reading covered everything. Apart from the works of his contemporaries, he had studied, with a remarkable objectivity, those of all the ancient and modern writers that were known in his time. This is witnessed by the historical references with which he enriched his treatise. Lagrange eliminated the contradictions and the inarticulateness which abounded in the work of his predecessors. He adopted the concepts and the postulates of the great creators of the previous century (Galileo, Huyghens, Newton). He surpassed Euler and d'Alembert. And he became preoccupied with the organisation of mechanics, the 1 The last edition to be published in LAGRANGE'S lifetime appeared in 1811. In the present book we have made use of the edition of 1853-1855, which was amended by Joseph BERTRAND and used certain manuscripts which had not been published during LAGRANGE'S life. LAGRANGE 333 foundation of its principles, the perfection of its mathematical language and the isolation of a general analytical method for solving its problems. His clarity of mind, his mathematical insight, served him so well that he arrived at an almost perfect codification of mechanics in the class ical field. In a detailed way, Lagrange made the following statement of his aims in an Avertissement. " To reduce the theory of mechanics, and the art of solving the associated problems, to general formulae, whose simple development provides all the equations necessary for the solution of each problem. " To unite, and present from one point of view, the different prin ciples which have, so far, been found to assist in the solution of problems in mechanics ; by showing their mutual dependence and making a judgement of their validity and scope possible. " As for the purely mathematical point of view which was Lagrange 's principal interest, he made the following declaration. " No diagrams will be found in this work. The methods that I explain in it require neither constructions nor geometrical or mechanical arguments, but only the algebraic operations inherent to a regular and uniform process. Those who love Analysis will, with joy, see mechanics become a new branch of it and will be grateful to me for thus having extended its field. " 2. LAGRANGE'S STATICS. In the historical part of his work Lagrange makes special mention of Archimedes, Stevin, Galileo and Huyghens. In his view, the equi librium of a straight and horizontal lever whose ends are loaded with equal weights and whose point of support is at the centre is " a truth that is evident on its own. " On the other hand, the principle of the superposition of equilibria, as fruitful as the principle of the super position of figures in geometry, is essential for a treatment of the angular lever. This leads to the principle of moments, in which connec tion Lagrange cites Guido Ubaldo. Lagrange refers to Stevin and to Galileo's mechanics in connection with the inclined plane. In the matter of the decomposition of a force into its components, he places Roberval before Stevin. To Lagrange, Descartes' principle and that of Torricelli were put forward without proof by their authors. Lagrange mentions Aristotle, Archimedes, Nicomedes and, among the moderns, Descartes, Wallis and Roberval, as having used the composition of motions. It was Galileo who had made first use of this concept in dynamics, in connection with the motion of projectiles. 334 THE PRINCIPLES OF CLASSICAL MECHANICS But, with good reason, Lagrange attributes the composition of forces, in the proper sense of the term, to Newton, Varignon and Lamy. An immediate connection, which Varignon saw and demonstrated by the theory of moments, exists between the principle of the lever and that of the composition of forces. Lagrange gives the following opinion on the justification of the rule of the parallelogram which had been given by Daniel Bernoulli. " By separating, in this way, the principle of the composition of forces from the principle of the composition of motion, the principal advantages of clarity and simplicity were lost, and the principle was reduced to being merely the result of geometrical constructions and analysis. " Lagrange then comes to the principle of virtual work, which he states in the following way. " Powers are in equilibrium when they are inversely proportional to their virtual velocities taken in their own directions." Lagrange mentions Guido Ubaldo as having been concerned in the formation of this principle. He refers to the concept of momento as used in Galileo's statics, recalls the part played by Descartes and Torricelli and honours Jean Bernoulli for having been the first to formulate the principle in all its generality. The justification of the principle of virtual work occupies a great deal of Lagrange's attention.1 " As for the nature of the principle of virtual velocities, it must be agreed that it is not sufficiently clear in itself to be formed into a first principle. But it can be regarded as the general expression of the laws of equilibrium, deduced from two principles [of the lever and of the composition of forces]. Further, in the demonstrations of this principle which have been given, it has always been made to depend on these by means which are more or less direct. But there is another general principle in statics which is independent of the principle of the lever and the principle of the composition of forces which, although it is customarily related to the others in mechanics, appears to be the natural foundation of the principle of virtual velocities — it can be called the principle of pulleys. " If several pulleys are mounted together on a single frame this assembly is called a polispaste or pulley-block. The combination of two pulley-blocks — one fixed and the other moveable — which is wound with a single string, one end of which is permanently attached and the other, acted upon by a power, forms a machine in which the power is to the weight carried by the moveable pulley-block as unity is to the number of strands which converge on this pulley-block ; this, 1 Mecanique analytigue, Vol. I, p. 21. LAGRANGE 335 if the strands are all supposed to be parallel and the friction and the stiffness of the strings is neglected. " By increasing the numbers of fixed and moveable pulley- blocks, and winding them all with the same string by means of various fixed and reversing pulleys, the same power, when it is applied to the moveable end, will be able to support as many weights as there are moveable pulley-blocks. Then, each weight will be to the power as the number of strands of the pulley-block support ing it is to unity. " For greater simplicity, make the last strand pass over a fixed pulley and let it support a weight instead of the power. We shall assume this weight to be unity. Also imagine that the different moveable pulley-blocks, instead of supporting weights, are attached to bodies — regarded as points — and arranged among each other so that they form any given system. In this way, by means of the string which is wound round all the pulley-blocks, the same weight will produce various powers, which act on the different parts of the system in the direction of the strings which converge on the pulley-blocks attached to these points. The powers will be to the weight as the number of strands is to unity. So that the powers themselves will be represented by the number of strands which come together and, by their tension, produce them. " Now it is clear that in order that the system drawn by these different powers may remain in equilibrium, it is necessary that the weight should be unable to descent by any infinitely small displacement of the points of the system. For since the weight always tends to descend, if there is any infinitely small displacement of the system which allows it to descend, it will necessarily do so and will produce this displa cement of the system. " Denote the infinitely small distances which this displacement would make the different points of the system travel by a, /?, y, . . . in the direction of the power which pulls them. Also denote the number of strands of the pulley-blocks applied at these points, to produce these powers, by P, $, JZ, . . . It can be seen that the distances a, /?, y, . . . will also be those by which the moveable pulley-blocks approach the associated fixed pulley-blocks. Further, it can be seen that these movements will decrease the length of the string which is wound round all the pulley-blocks by the quantities Pa, ()/9, fty, ... So that, because of the fixed length of the string, the weight will descend throughout the distance P« + QB + Ry + . . . 336 THE PRINCIPLES OF CLASSICAL MECHANICS " Therefore, in order that the powers represented by the numbers P, $, JR, . . . may be in equilibrium, it will be necessary that the equation Pa + Q$ + Ry + . . . = 0 should obtain. This is the analytic expression of the general principle of virtual velocities. " We remark here, with Jouguet, l that Lagrange's demonstration is based on physical facts — on certain simple properties of pulleys and strings. Lagrange also assumes the truth of the principle in a very particular case, which reduces to the hypothesis of Huyghens and Torricelli. We owe to Lagrange the elegant method called that of multipliers. The object of this was to express, in a general way, the problems of statics by means of mathematical equations.2 Lagrange expressed the constraints of the system by equations of the type L = 0 M= 0 N = 0 ... where L, M, N are finite functions of the coordinates of the points of the system. Differentiating these conditions, Lagrange writes dL = 0 AM = 0 dN = 0 . . . (He does not exclude equations of constraint between differentials that are not " exact differences " — these are the constraints that are now called non-holonomic.) Lagrange declares, " These equations should only be used to elim inate a similar number of differentials in the general formula of equi librium, after which the coefficients of the remaining differentials all become equal to zero. It is not difficult to show, by the theory of the elimination of linear equations, that the same result will obtain if the* various equations of condition dL = 0, dM = 0, dN = 0, . . . are each multiplied by an indeterminate coefficient and simply added to the equation concerned ; if then, the sum of all the terms which are multiplied by the same differential are equated to zero, which will give as many particular equations as there are differentials ; and if, finally, the indeterminate coefficients by which the equations of con- 1 L. M., Vol. II, p. 179. 2 Mecanique analytique. Vol. I, p. 69 et seq. LAGRANGE 337 dition have been multiplied are eliminated from the last set of equation. " Whence the rule stated by Lagrange for finding the conditions of equilibrium of any system — " The sum of the moments [that is, apart from sign, the virtual works] of all the powers which are in equilibrium will be taken, and the differential functions which become zero because of the conditions of the problem will be added to it, after each of these functions has been multiplied by an indeterminate coefficient ; then the whole will be equated to zero. Thus will be obtained a differential equation which will be treated as an ordinary equation of maximis et minimis. From this will be deduced as many equations as there are variables. These equations, being then rid of the indeterminate coefficients by elimination, will provide all the conditions necessary for equilibrium. " The differential equation concerned will therefore be of the form Pdp + Qdq + Rdr + ... + UL + pdM + vdN + . . . = 0 in which A, /*, v are the indeterminate quantities. In the sequel we shall call this the general equation of the equilibrium. " Corresponding to each coordinate of each body of the system, such as x, this equation will give an equation of the form P3? , n 39 , P dr _L . i 9L t BM . dN L n Pf- + Q^ + R^~ + ... + A- -- h/*-a -- h V — +...= 0. dx ox dx dx ^ dx ox Therefore the number of these equations will be equal to the number of all the coordinates of all the bodies. We shall call these the particular equations of the equilibrium. " It only remains to eliminate the multipliers A, u, v. Taking account of the equations of constraint, the problem of the determination of the coordinates of the different elements of the system is thus solved. Lagrange did not confine himself to this abstract analysis, but gave it a physical interpretation. The terms AdL, /idM, vdN " must be regarded as representing the moments [of virtual works] of certain forces applied to a system. " Thus dL is written in the form dL (*', /, *', *", /', *"...) = dL' + *L"+ ... In this equation (x, y\ *'), (#", y", 2"), etc. . . . represent the coor dinates of each particle, and dl/, dL' \ etc. . . . only depend on (x\y\ zr), (x", y", z"), etc. . . . respectively. Lagrange then verifies that the term hdL is equivalent to the effect of different forces 338 THE PRINCIPLES OF CLASSICAL MECHANICS applied, respectively, at the points (#', y', z'), (#", y", z"), etc. . . . and normal to the different surfaces defined by the equation dL = 0. In this equation the variation is first performed with respect to (#', y', 2'), then with respect to (#", y", 2"), etc. . . . Lagrange concludes, " It follows from this that each equation of condition is equivalent to one or more forces applied to the system in given directions. So that the state of equilibrium of the system will be the same whether the consideration of forces is used, or whether the equations of condition themselves are used. " Conversely, these forces must take the place of the equations of condition resulting from the nature of the given system, so that by making use of these equations it will be possible to regard the bodies as entirely free and without any restraint. And from this is seen the metaphysical reason why the introduction of the terms Ad£ -f judM + ... in the general equation of equilibrium ensures that this equation can then be treated as if all the bodies were entirely free. . . . " Strictly speaking, the forces in equation take the place of the resistances that the bodies would suffer because of their mutual con straint or because of obstacles which, by the nature of the system, could oppose their motion ; or rather, these forces are merely the same forces as the resit ances, which are equal and directly opposite to the pressures exerted by the bodies. As is seen, our method provides a means of determining these forces and resistances. . . . " The considerable progress achieved by Lagrange in the analytical application of the principle of virtual work is very evident. Lagrange does not become inordinately eloquent on the concept of force itself. He confines himself to saying, " By force or power is understood, in general, the cause which imparts, or tends to impart, motion to the bodies to which it is supposed to be applied ; further, it is by the quantity of motion imparted, or which may be imparted, that the force must be represented. In the state of equilibrium the force does not have actual effect ; it only provides a tendency to motion. But it can always be measured by the effect that it would produce if it were not arrested. " l 1 M&anique analytique^ Vol. I, p. 1. LAGRANGE 339 3. LAGRANGE AND THE HISTORY OF DYNAMICS. In Lagrange's view, dynamics is " the science of accelerating or retarding forces and the varying motions which they must produce. This science we owe entirely to the moderns, and Galileo is the one who laid its first foundations. . . . Huyghens, who seems to have been destined to perfect and complete most of Galileo's discoveries, sup plemented the theory of heavy bodies by the theories of the motion of pendulums and centrifugal forces, and thus prepared the way for the great discovery of universal gravitation. Mechanics became a new science in the hands of Newton, and his Principia, which appeared in 1687, was the occasion of this revolution. " I Thus, neglecting all the vicissitudes of Aristotelian mechanics and the few inspirations of the Schoolmen, Lagrange acknowledged a century of evolution in the subject that he was to codify. Lagrange ascribes the two principles of the force of inertia (that is, inertia) and the composition of motions to Galileo. He analyses the method followed by Huyghens in his work on the centrifugal force in the following way. " For the estimation of forces, it suffices to consider the motion produced in any time, finite or infinite, provided that the force may be regarded as constant during this time. Consequently, whatever the motion of the body and the law of acceleration may be, since, by the properties of the differential calculus the action of every accele rating force may be regarded as constant during an infinitely small time, it will always be possible to find the value of the force which acts on the body at each instant. This is done by comparing the velocity produced in this instant with the duration of the same instant ; or by comparing the distance which the body travels with the square of the duration of the same instant. It is not necessary, even, that the distance should be actually travelled by the body, it is sufficient that it may be supposed to have been travelled by a compound motion, since the effect of the force is the same in one case as in the other. " 2 In a careful analysis of the use of mathematics, Lagrange remarks that " Newton made constant use of the geometric method as simplified by the consideration of the first and last ratios. " Euler's Mechanica (1736) is, to Lagrange, the first great work in which Analysis was applied to the science of motion. As for MacLaurin's Treatise on Fluxions (1742), this was the first work which systematically used 1 Mfaanique analytique, Vol. I, p. 207. 2 Ibid., p. 210. 340 THE PRINCIPLES OF CLASSICAL MECHANICS the rectangular components of the force instead of their tangential and normal components. Lagrange then comes to a principle which allows the determination of the force on bodies in motion, having regard to their mass and velocity. 66 This principle consists in that, in order to impart to a given mass a certain velocity in some direction, whether the mass be at rest or in motion, the necessary force is proportional to the product of the mass and the velocity and its direction is the same as that of the velocity. " l Here Lagrange cites Descartes as having first realised the existence of this principle, but as having deduced from it incorrect rules about the impact of bodies. On the other hand, Wallis made successful use of the principle to discover the laws of the transfer of motion in the impact of hard or elastic bodies. And Lagrange continues, " Just as the product of the mass and the velocity represents the finite force of a body in motion, so the product of the mass and the accelerating force — which we have seen to be represented by the element of velocity divided by the element of time — will represent the elementary or nascent force. Ajid this quantity, if it is considered as the measure of the effect that the body can exert because of the velocity which it has assumed, or which it tends to assume, constitutes what is called pressure ; but if it is regarded as a measure of the force or power necessary to impart this same velocity, it is then what is called motive force. " In modern language, the finite force of a body in motion is represent ed by the product mv, and the " elementary or nascent force " u * bym5T Lagrange does not openly take sides between Euler's thesis — based on the law Fdt = mdv (where F is the static force) — and d'Alem- bert's thesis. This matter of principle interested him less than the formal organisation of dynamics, which was the primary object of his own treatise. Because of the work of his predecessors, the mechanics of a particle had no mystery for him. Primarily, he sought to provide statics, and then the dynamics of systems, with the general method that they still lacked. In turn, Lagrange analyses the four principles of dynamics — the conservation of living forces ; the conservation of the motion of the centre of gravity ; the conservation of moments or the principle of areas ; and the principle of the least quantity of action. Lagrange says, legitimately, that the first of these principles goes back to Huyghens " in a form a little different from that in which it 1 Mfaanique analytique, Vol. I, p. 213. LAGRANGE 341 is presented now. " Jean Bernoulli, following Leibniz, fashioned it into the principle of the conservation of living forces. Daniel Bernoulli, after applying it to fluids, extended it (in the Memoires of Berlin for 1748) to a system of bodies attracting each other, or tending towards fixed centres, according to any law which is a function of distance. The second principle is due to Newton and was revived by d'Alem- bert. The third principle, discovered by Euler,1 Daniel Bernoulli,2 and d'Arcy,3 is only the generalisation of a theorem of Newton concerning several particles attracted by the same centre. D'Arcy went further and sought to make the principle of areas into a principle of the conservation of action. Lagrange protests, 66 As if this vague and arbitrary nomenclature were the essence of the laws of nature and could, by some secret property, elevate the simple results of the known laws of mechanics into final causes. " 4 The criticism which Lagrange directs against Maupertuis' principle merits quotation. " Finally I come to the fourth principle, which I call that of least action by analogy with that which Maupertuis gave under the same name, and which the writings of many illustrious authors have since made so well-known. This principle, looked at analytically, consists in that, in the motion of bodies which act upon each other, the sum of the products of the masses with the velocities and with the distances travelled is a minimum. The author deduced from it the laws of the reflection and refraction of light, as well as those of the impact of bodies. " But these applications are too particular to be used for esta blishing the truth of a general principle. Besides, they have a somewhat vague and arbitrary character, which can only render the conclusions that might have been deduced from the true correctness of the principle unsure. Further, it seems to me that it would be wrong to place this principle, presented in this way, among those which we have just given. But there is another way in which it may be regarded, more general, more rigorous, and which itself merits the attention of the geometers. Euler gave the first hint of this at the end of his Traite des isoperimetres, printed at Lausanne in 1744. He demon strated, in the trajectories described under the action of central forces, that the integral of the velocity multiplied by the element of the curve 1 Opuscules, Vol. I, 1746. 2 Memoires de Berlin, 1746. 3 Memoires de FAcademie des Sciences, 1747. 4 Mecanique analytigue, Vol. I, p. 228. 342 THE PRINCIPLES OF CLASSICAL MECHANICS is always a maximum or a minimum. By means of the conservation of living forces I have extended this property, which Euler discovered in the motion of isolated bodies and which seemed confined to these bodies, to the motion of any system of bodies which interact in any way. From this has come a new general principle, that the sum of the products of the masses with the integrals of the velocities, each of which is multiplied by the element of distance travelled, is invariably a maximum or a minimum. " This is the principle which I now give, however improperly, the name of least action. I regard it not as a metaphysical principle, but as a simple and general result of the laws of mechanics. " x 4. LAGRANGE 's EQUATIONS, Lagrange was able to put the equations of dynamics into a very general and valuable form which has now become classical. For each element, of mass m, of a system, Lagrange defines " the forces parallel to the axes of coordinates which are used, directly, to move it, " to be d*x d*y d*z ~ He regards each element of the system as acted upon by similar forces, and concludes that the sum of the moments 2 of these forces must always be equal to the sum of the given accelerating forces which act on each element. Thus he writes Rdr + ...) = 0 the given forces P, (), JR, . . . being supposed to act on each element along the lines p, £, r, . . . Lagrange transforms the first sum by using the identity d*xdx + d*ydy + dzzdz = d (dxdx + dydy + dzdz) — -6 (dx* + dy* + dz*}. By a change of variables in which each differential dx, dy, dz, . . . is expressed as a linear function of the differentials d£, dip? d<p, . . . , 1 Mecanique analytigue, Vol. I, pp. 229, 230. 2 In the sense already encountered in LAGRANGE'S statics. IAGRANGE 343 Lagrange establishes that if 0 is the transform of the quantity i (da* + dy* + dz*) £ then the following equation is identically true. Lagrange confines himself to forces P, Q, R, . . . for which the quantity Pdp + Qdq + Rdr + is integrable, which, he declares, " is probably true in nature. " This enables him to suppose that Sm (Pdq + Qdq + Rdr +...) = d SroZT(f, y9 (p. . .). The general equations of dynamics are then written in the form sdf + y% + . . . - o by putting ~,3T dT dV with * d* dzz Having arrived at these results, Lagrange examines the particularly interesting circumstance in which the variables f, yj, . . . are exactly sufficient to characterise the motion of the system after all the equations of constraint have been eliminated. 66 If, in the choice of the new variables £, ^, . . . , regard has been paid to the equations of condition provided by the nature of the proposed system, so that the variations are now completely independent of each other and that, consequently, their variations <5£, <5y>, . . . , remain absolutely indeterminate, then the particular equations will serve to determine the motion of the system, since these equations are equal in number to the variables £, ^, . . . on which the position of the system at each instant depends. " 1 1 Mecanique analytique, Vol. I, p. 291. 344 THE PRINCIPLES OF CLASSICAL MECHANICS Lagrange connects this analysis with the method of multipliers which he introduced in statics. If the variables f, ip, . . . are greater in number than the degrees of freedom of the system, they will be related by the equations L = 0 M= 0 N= 0 ... Then Lagrange's general formula becomes Sdg + Wdy + ... + 16L + pdM + vdN + . . . = 0 whence the equations of motion ^ dL dM dN aL BM t -J- A. ~r— -p U -r — • oip oyj which must be associated with the equations of constraint. The method of multipliers, which Lagrange himself only applied here to the systems of constraints which are now called holonomic, is easily extended to non-holonomic constraints — that is, to constraints which cannot be expressed finitely as functions of f, ip, . . . 5. THE CONSERVATION OF LIVING FORCES AS A COROLLARY OF LAGRANGE *S EQUATIONS. Better than d'Alembert had been able to do, Lagrange established that the conservation of living forces is a consequence of the equations of dynamics, as long as the constraints are without friction and independent of time. For this purpose, Lagrange considers the true motion of the system between the time t and the time t + dt ; that is, he substitutes dx, Ay, dz, .. . and dp, dq, dr, . . . for dx, 6y, 6z, . . . and dp, dq, dr, . . . in the general formula. This enables him to write Q fdxd2x H U77I If the quantity Pdp + Qdq + Rdr + ... is integrable, then LAGRANGE 345 " This equation includes the principle known by the name of the conservation of living forces. Indeed, since dx* + dy2 + && is the square of the distance which the body travels in the time eft, then x -. — — — ! — — will be the square of the velocity and m ~ dt2 dtr will be its living force. Therefore c fdx* + dy2 + • 3? 771 will be the living force of the whole system, and it is seen, by means of the equation concerned, that this living force is equal to the cpian- tity 2H — 2 Slim, which only depends on the accelerating forces which act on the bodies and not on their mutual constraints. So that the living force is always the same as that which the bodies would have acquired if they had moved freely, each along the line that it described, under the influence of the same powers. " z Thus Lagrange discovers the same principle as that formulated by Huyghens to be a simple corollary of his general equations. 6. THE PRINCIPLE OF LEAST ACTION AS A COROLLARY OF LAGRANGE's EQUATIONS. Lagrange starts from the equation of living forces d#2 + dy2 + A and differentiates it to obtain STTI (udu + dll) = 0 or Sm (Pdp + Qdq + Rdr +...) = — Smudu. Substitution in the general formula leads to or Sm 7^; u2 -j- — udu = 0 Mecanique analytique, Vol. I, p. 268. 346 THE PRINCIPLES OF CLASSICAL MECHANICS or again [ *(*** dt and finally oC f 7 c fdx . dy dz \ oom uds = om I -=- ox + -=r- or 4- -=- oz . J \<ft { dt J dt / If it is supposed that the variations <5#, <5y, dz are zero at the ends of the ranges of integration, then <5S J muds = 0, Lagrange concludes, " In the general motion of any system of bodies, actuated by mutual forces of attraction, or by attractions towards fixed centres which are proportional to any function of the distance, the curves described by the different bodies, and their velocities, are necessarily such that the sum of the products of each mass by the integral of the product of the velocity and the element of the curve is necessarily a maximum or a minimum ; provided that the first and last points of each curve are regarded as fixed, so that the velocities of the corresponding coordinates at those points are zero. " l Maupertuis' principle is thus found to be valid, in a more general form than that which Euler gave it. Moreover, this principle expresses the extremal character of the living force between two known confi gurations of the system. This Lagrange establishes in the following way. " Since ds — udt, the formula Sm uds which is either a maximum or a minimum, can be put in the form Sm u2dt or eftSmu2. Here Smw2 represents the living force of the whole system at any time. Thus the principle reduces to — the sum of the instantaneous living forces of all the bodies, from the moment that they start from given points to that when they arrive at other given points, is a maxi mum or a minimum. It could be called, with more justice, the prin ciple of the greatest or least living force, and this way of regarding it would have the advantage of being general, since the living force of a system is always greatest or least in the equilibrium condition. " 2 1 Mtcanique analytique, Vol. I, p. 276. 2 Ibid., p. 281. LAGRANGE 347 7. ON SOME PROBLEMS TREATED IN THE " MECANIQTJE ANALYTIQUE." Mecanique analytique includes a study of a great number of pro blems which we are not able to treat in this book. We note, however, that Lagrange initiated a general method of approximation in dynamical problems which was based on the variation of arbitrary constants ; that he developed the theory of small motions ; that he studied the stabilty of equilibrium, and stated that equilibrium is necessarily stable when the potential of the given forces is a minimum. This demonstration was to be perfected by Lejeune-Dirichlet. Lagrange also studied in detail the motion of a heavy solid of revolution which was suspended from a point on its axis, and expressed the solution in terms of elliptic integrals. 8. LAGRANGE'S HYDRODYNAMICS, After describing the historical development of hydrodynamics, Lagrange made a very important contribution to the subject. As a supplement to Euler's variables, Lagrange introduced the variable with which his name is still associated into the kinematics of continuous media. The actual coordinates of an element of the medium are considered as functions of the time and of the initial coordinates a, 6, c, of the same element. Lagrange established a fundamental theorem on the permanence of the irrotational property in fluid motion. If the fluid is first supposed to be incompressible and homogeneous, and its density is taken equal to unity, Lagrange1 also assumed that the accelerating forces X, Y, Z which act on the elements of the fluid are such that Xdx + Ydy + Zd* is an exact differential dV. Lagrange writes fdu du du , du dv , dv s-x+v^ Sw , dw The right-hand side of this equation, like the left hand side, must be an exact differential Now the right-hand side can be written as 1 Mfaanique analytique, Vol. II, p. 268. 348 THE PRINCIPLES OF CLASSICAL MECHANICS Lagrange remarks that this quantity will be an exact differential whenever udx + vdy + wdz is such, " but as this is only a special supposition, it is necessary to inquire in what cases it can and must be appropriate. " Lagrange then verifies that when u, v, w are expanded as functions of the time, in the form V = . . . W = it is necessary that, whenever udx + v'dy + w'dz is an exact differential, that u"dx + v"dy + w"<fc, u'"dx + v'"dy + w'"dx, etc should also be exact differentials. He concludes, " From this it follows that if the quantity udx + vdy + wdz is a total exact differential when t = 0, it must also be a total exact differential when t has any other value. Therefore, in general, since the origin of t is arbitrary, and since t can either be taken positive or negative, it follows that if the quantity udx + vdy + wdz is a total exact differential at any time, it must be such at all other times. " Accordingly, if there is a single instant at which it is not a total exact differential, it can never become such throughout the motion. For if it were a total exact differential at any instant, it would also be such at the first. " This theorem of Lagrange is a fine example of discovery achieved by a procedure which appears to be purely mathematical. In sympathy with the spirit of the time, which readily assumed that nature conformed to simple laws, Lagrange declared " that it is possible to ask whether there are motions for which udx + vdy + wdz is not a total exact differential. " To answer this question, he shows that in the motion u = gy v = — • gx w = 0 the condition that udx + vdy + wdz should be a complete differential is not satisfied, although it is possible to write p = V- — (*2 + y2) + funct. t. 2t Now " it is clear that these values of u, v, w represent the motion of a fluid which rotates with a constant angular velocity equal to g about the fixed axis of coordinates z. And it is known that such a motion can always take place in a fluid. From this it can be concluded that in the oscillations of the sea due to the attraction of the Sun and LAGRANGE 349 the Moon, it cannot be supposed that the cpiantity udx + vdy -j- is integrable, since it is not so when the fluid is at rest with respect to the Earth and only has the same rotational motion as the Earth. " Lagrange extended his theorem to compressible fluids by intro ducing an " elasticity " that was a function of density alone, so that — is the differential of some function E(Q). e We also mention Lagrange's study of the motion of a fluid in an almost horizontal shallow canal. This motion is governed by an equation similar to the equation of the propagation of sound. The wave velocity turns out to be proportional to the s<juare root of the depth of the fluid if the canal has a uniform breadth. PART FOUR SOME CHARACTERISTIC FEATURES OF THE EVOLUTION OF CLASSICAL MECHANICS AFTER LAGRANGE FOREWORD It would seem that the valuable function of history is that analysing the paths which scientific thought has travelled in the creation, in a limited field like that of classical mechanics, of a rationally organised structure. The material for this study thus consist of the vicissitudes en countered on these paths, the interaction between currents of thought which were in principle divergent, or even opposed. It is difficult material, sometimes deceptive, but always revealing of the profound difficulties of research and — for this reason — instructive. After Lagrange, after the efforts of the students of the XVIIIth Century, mechanics had attained this rational structure. It lasted until the impact of the needs of relativistic and quantum physics. The intervening period was a didactic one, which we are reluctant to deal with for fear of duplicating the books that have, rightly, become classical. That is why, in the following pages, we shall confine ourselves to some characteristic features of the evolution of classical mechanics after Lagrange. We shall be concerned with discussions of the prin ciples themselves, and with certain isolated facts to which it seems natural to attach a historical significance because of their influence on the later development of mechanics. 23 CHAPTER ONE LAPLACE'S MECHANICS (1799) 1. LAPLACE AND THE PRINCIPLES OF DYNAMICS. We shall discuss here only that part of Laplace's work which is directly concerned with the principles of dynamics. Laplace referred the motion of bodies to " an infinite space, at rest and penetrable to matter. " x The concept of force evoked the following comment. " The mechanism of that remarkable agency, force, by which a body is moved from one place to another, is and always will be unknown. It is only possible to determine the laws which govern its behaviour. A force acting on a particle will necessarily set it in motion, if there is nothing to prevent this. The direction of the force is that of the straight line which the particle is made to describe. " 2 Laplace defined inertia as the tendency of matter to remain in its state of rest or motion. That the direction of motion was constant appeared obvious to him ; with respect to its uniformity, he pointed out that the law of inertia was the simplest conceivable law, and that it was justified by astronomical and terrestrial observations. The Laplace endeavoured to prove that " force " is proportional to the velocity. He is concerned here, of course, with the force of a moving body. Let v be the velocity of the Earth, common to all bodies on its surface, and / be the force which a particle M experiences because of this motion. The ration- is an unknown function of /, say <p(f). The form of this function has to be found by a method which has recourse to experimental observation. Suppose that M is also acted upon by another force, /', which 1 Mecanique celeste, Part I, Book I, p. 3 (1799). 2 Ibid., p. 4. LAPIACE 355 combines with f to form a resultant F according to the parallelogram rule. Under these conditions the particle will acquire a certain velocity U. Laplace now argued that /' could be considered as " infinitely small " compared with /. " The greatest forces that we are able to impress on bodies on the surface of the earth are much smaller than those that it experiences because of the motion of the earth. " Accordingly, the relation yields J if /'2 is neglected in comparison with /. It follows that g> (F) = <p (/) + f-f <p'(f) with ?'(/) = & J dt The relative velocity of M with respect to the Earth, U — v, equal to F,p(F)-f<p(f) which can easily be shown to lead to the equation is From this it follows, in the general case in which the directions of f and f are not the same, that this relative velocity must have a compo nent perpendicular to that of the impressed force/' unless <p'(f) vanishes. (It is assumed that the scalar product / • /' is not zero.) At this point Laplace appeals to experiment. " Thus, imagine that a sphere at rest on a smooth horizontal plane is hit by the base of a right cylinder, moving along the direction of its axis which is supposed to be horizontal. The apparent relative motion of the sphere should not be parallel to this axis, for all positions of the axis with respect to the horizon. Here is a simple means of finding out by experiment whether <p'(f) has an appreciable value on the Earth. But the most accurate experiments do not demonstrate any deviation of the apparent motion of the sphere from the direction of the impressed force. From which it follows that on the Earth <p'(f) is very nearly equal to zero. Its value, however inappreciable it might be, would make itself apparent in the length of the oscillations 356 THE EVOLUTION OF CLASSICAL of a pendulum, which would vary according to the position of the plane of its motion with respect to the direction of the Earth's motion. The most accurate observations do not reveal any such difference. We must then conclude that <p'(f) is inappreciable, and can be supposed to be zero, on the Earth. " If the equation q>'(f) = 0 was obtained whatever the force /might be, <p(f) would be constant and the velocity would be proportional to the force. It would also be proportional to the force if the function <p(f) were only composed of a single term, since otherwise V'(f) would never be zero. Therefore, if the velocity were not propor tional to the force, it would be necessary to suppose that in nature the function of the velocity which represents the force consists of several terms, which is unlikely. It would also be necessary to suppose that the velocity of the Earth is exactly that which the equation <p'(f) — 0 requires, which is against all probability. Moreover, the velocity of the Earth varies in the different seasons of the year — it is about a thirtieth greater in winter than in summer. This variation is still more considerable if, as everything seems to show, the solar system is in motion through space — for whether this progressive motion combines with that of the Earth or whether it is opposed to it, during the course of the year there must result large variations in the absolute motion of the Earth. This would modify the equation concerned and thus the relation of the impressed force to the absolute velocity, if this equation and this velocity were not independent of the motion of the Earth. However, observation has not revealed any appreciable variation. " 1 Laplace concludes, " Here then are two laws of motion ; namely, the law of inertia and that according to which the force is proportional to the velocity, and these are provided by observation. They are the simplest and most natural that could be imagined and undoubtedly they derive from the nature of matter itself. But since this nature is unknown they are merely, for us, observed facts ; moreover, the only ones which mechanics borrows from observation. " 2 Laplace next gave his attention to " forces which appear to act in a continuous manner, like gravity. " Here, like Carnot, Laplace considers that gravity acts in successive impulses at infinitely small intervals of time. " We suppose that the interval of time which separates two actions of some force is equal to the element of time dt. It is clear [that the instantaneous action of the force] must be supposed to be proportional to the intensity and to the element of time in which 1 Mecanique celeste, Part I, Book I, p. 17. 2 Ibid., p. 18. LAPLACE 357 it is supposed to act. Thus, representing the intensity by P, it must be supposed, at the beginning of each instant A, that the particle is actuated by a force Pdt and that it is moved uniformly during that instant. " l 2. THE GENERAL MECHANICS COMPATIBLE WITH AN ARBITRARY RELATION BETWEEN THE " FORCE " AND THE VELOCITY. In the last § we have seen Laplace emphasise the communication of motion without seeking to elucidate the original causes of the motion. He thus belongs to the tradition of d'Alembert and Carnot. In parti cular, like Carnot, he asserted the experimental character of the laws of mechanics. His analysis — this time entirely original — of the notion of the force of a body in motion led him, by an innate propensity to purely mathematical generalisation, to an extension of dynamics which encompassed the ideas that the physicists were to use, a century later, in special relativity. This extension was the subject of Chapter VI of the first part of Book I of the Mecanique celeste. The chapter is called " On the laws of motion of a system of bodies associated with all possible mathemat ical relationships between the force and the velocity. " Laplace remarks (as we have mentioned in the preceding paragraph) that there is an infinite number of self- consistent ways of expressing the " force " in terms of the velocity. This infinity corresponds to all possible forms of the relation between the force and the velocity. F=<p(v) The general equation of the dynamics of systems is and is valid when the " force " is proportional to the velocity. In order to obtain the generalisation which is sought, it is sufficient to assume that the body of mass m is actuated by a " force " whose components parallel to the axes are doc dy dz *w2s *(v]Ts *(v]ds- In the instant following this force becomes . . dx , * / , . dx\ 9(t))_ + d^(v)._j etc. 1 Mtcanique celeste. Part I, Book I, p. 19. 358 THE EVOLUTION OF CLASSICAL , . dx f^dTS + Then the general equation of the dynamics of systems takes the form *)-"*)}-* Here — , -j- and -— appear as products with the function — — , which at at at v reduces to unity " for that natural law according to which the force is proportional to the velocity. " Laplace remarked that this difference makes the solution of the problems of mechanics very difficult. But the principle of the conservation of living forces, the principle of areas and the principles of the centre of gravity and of least action, can be extended to this case. The extension of the principle of living forces is obtained by sub stituting dx9 dy and dz for 6x, dy and dz in the general equation. Thus, if U is the function of the forces, £ J mvdv-(p'(v) =U + h with <p'(v) = " The principle of the conservation of living forces therefore obtains for all the possible mathematical relationships between the force and the velocity, provided that the living force of a body is understood as the product of its mass and twice the integral of the product of its velocity and the differential of the function of velocity which represents the force. " In the same way, Laplace extended the theorem of quantities of motion to an isolated system. He generalised the principle of areas to the form fxdy — ydx\ cp (v) m J J — . TJU. — constant. \ dt ) v Finally, he wrote the generalised principle of least action as Thus, as early as 1799, Laplace was able to formulate the general mechanics of which the dynamics of special relativity, in a given LAPLACE 359 reference system, is only a particular case. However, there is a slight difference of meaning between Laplace's purely mathematical concep tion and that of the modern physicists. To Laplace, the mass m of a particle remained constant and it was the momentum which was no longer proportional to the velocity. In the physical theory of relativity, on the other hand, the mass M becomes a function of the velocity while the momentum remains in the form Mv. In order to pass from one of these systems to the other, it is suffi cient to put cp(v) M = m • v 3. LAPLACE AND THE SIGNIFICANCE OF THE LAW OF UNIVERSAL GRAVITATION. In his Exposition du Systeme du Monde Laplace recalls that when Newton formulated the principle of universal gravitation, Descartes had precisely managed to " substitute the intelligible ideas of motion, impulse and centrifugal force for the occult quantities of the Aris totelians. " 1 His system of vortices met with the approval of the philosophers, " who rejected the obscure and meaningless doctrines of the Schoolmen, and who believed that they saw those occult features, that French philosophy had so legitimately banned, reborn in the attractions. " 2 Laplace opined that Newton would have deserved this reproach if he had been content to attribute the elliptic motion of the planets and comets, the inequalities of the motion of the Moon, of the terres trial degrees of latitude and gravity, to the universal attraction without showing the connection between this principle and these phenomena. But the geometers, rectifying and generalising Newton's demon strations, had been able to verify the perfect agreement between the observations and the results of the analysis. Laplace regarded " this analytical connection of particular facts with a general fact " as a properly constituted theory. And he flattered himself with having obtained one in the deduction on the effects of capillarity from a short range interaction between molecules ; a true theory, one which expresses the rigorous agreement of the calculation and the phenomena. Here we see portrayed the dogma of universal attraction. However — and this is essential — it is deprived of the a priori character of the 1 P. 377. 2 P. S78. 360 THE EVOLUTION OF CLASSICAL assertion of a quality in the sense that the Schoolmen used. Although it must certainly have passed through his mind, Laplace did not use the term dogma in this connection. For his attitude, supported at many points by experiment, was not necessarily of a dogmatic character. Laplace assumed, moreover, that the following question could be asked — " Is the principle of universal gravity a primordial law of nature, or is it merely the general effect of an unknown cause ? " l Laplace also asked whether the propagation of the attraction was instantaneous. An attempt to explain the secular acceleration of the Moon's motion had led him to assume that, if the velocity of propagation was finite, it must be seven million times greater than that of light. . . . Thus he declared for an instantaneous propagation. Further, he wrote, " Doubtless the simplicity of the laws of nature must not necessarily be judged by the ease with which we appreciate them. But since those that seem most simple to us agree perfectly with all the phenomena, we are well justified in regarding them as being rigorous. " We see that Laplace's attitude was a moderate one, and that, to him, the certainty of a natural law depended on a kind of passage to the limit in the mathematical sense of the term. 1 P. 384. CHAPTER TWO FOURIER AND THE PRINCIPLE OF VIRTUAL WORKS (1798) We owe to Fourier a demonstration of the principle of virtual works which is based on the equilibrium of the lever, and which has been used as the basis of the presentations of the principle that are now classical.1 Fourier borrowed the notion of virtual velocity from Jean Bernoulli.2 On the other hand, he called the moment of a force the product (change of sign) of this force with the virtual velocity of the point to which it is applied. If a point is in equilibrium under the action of n forces, Fourier first verifies that the total moment of these forces is zero for an arbi trary displacement of the point. He then seeks the total moment of two equal and opposed forces " applied at the ends of a straight inflexible line " and acting in the direction of this line. " If, at first, the two points at which these forces act are regarded as entirely free, and if each of the points is taken as the fixed centre of the force which acts on the other, it will be easy to see that, since the distance apart of the points is a function of their coordinates, the virtual velocity of the first will be equal to differential of the distance when the variation is made with respect to the coordinates of this point alone. It will be the same for the second point. So that the total moment, which is proportional to the sum of the virtual velocities, will also be proportional to the sum of the partial differentials which represent these velocities — that is, proportional to the complete dif ferential of the distance between the two points. Thus the total moment of the two forces is zero if the distance between the two points is constant. If the two forces are repulsive, the total moment is negative when 1 Memoire sur la Statique, contenant la demonstration du principe des vitesses vir- tuelles et la theorie des moments, Journal de VBcole poly-technique, 5th cakier, 1798. 2 See above, p. 232. 362 THE EVOLUTION OF CLASSICAL the distance between the two points increases and positive when this distance decreases. Converse results are obtained when the forces are attractive. Fourier then considers " two inflexible (and perfectly smooth) surfaces which resist each other " and studies the total moment of their mutual reactions in any disturbance of the system. He considers two points on the common normal to the surfaces at the point of con tact and such that one lies inside each surface. These two points cannot be closer together than they are in the equilibrium position. So that either their distance apart increases or it does not change when the system is disturbed. " The first distance is the smallest of all those which occur when the position of the two surfaces is varied in such a way that they remain in contact with each other. Since the law of continuity is obeyed, it is necessary that the differential should be zero. " Since the total moment of the reactions is proportional to the variation of the distance of the two points at which they act, it therefore remains zero, as long as the surfaces remain in contact, whatever the displacement may be. In order to generalise these results Fourier observes that the mo ments combine and decompose like forces (if a solid body is concerned). If a solid body is considered to be in equilibrium under the action of n forces it is established that the total moment of the n forces is necessarily zero. The converse is also true. Fourier then imagines a system of bodies to be connected by in- extensible threads and acted upon by any forces which are such that there would be equilibrium independently of any external resistance. The forces which act on each body cancel each other out. Apart from the forces directly applied to the body, these forces comprise the tension of the threads between points of this body and points of neighbouring bodies. " That is why, in considering simultaneously all the forces which act on all the bodies, their total moment can be said to be zero for all conceivable disturbances — even for those which the presence of the threads does not allow. It is now necessary to select, from among these disturbances, those which satisfy the equations of condition ; and, for these particular disturbances, to discover the value of the total moment of those forces which are due to the tensions alone. " This value is zero. For each of the threads is acted upon by two equal and opposed forces, and the distance between the extremities is constant. From this it follows that the total moment of the applied forces alone is also zero. If the distance between the ends of the threads does not remain FOURIER 363 constant, it can only become smaller. Since the forces of tension tend to decrease this distance, the total moment of the forces of tension is negative. Therefore, the sum of the moments of the applied forces alone can only be positive for disturbances of this kind. Fourier next considers an " undefined assembly of hard bodies " whose shapes and dimensions are arbitrary and which are supported upon each other. Each body is in equilibrium under the action of the forces which are applied to it and the resistances of neighbouring bodies. If two neighbouring bodies always remain in contact during the disturbance of the system, albeit at different points, the moment of their reactions is zero. It is negative if the bodies happen to separate. ** In considering the combination of all the forces which act on all the bodies, it is certain that some of the moments must be zero for all the disturbances which can be imagined — even for those which may be prevented by the mutual impenetrability of the solids. Now, for displacements compatible with the latter condition, the moment of all the forces of pressure is either zero or negative. Therefore, for all the possible disturbances, the sum of the moments of the applied forces alone is either zero or positive l — it is zero when the equations which express the condition that contact must take place are satisfied, and positive whenever two bodies which touch each other, or act upon each other, become entirely separated. There is no possible disturbance for which the sum of the moments can be negative." Fourier treats incompressible fluids by considering that their different points are subject to an interaction which opposes every variation of the distances between the points. He then proceeds to the logical reduction of the theorem of virtual work to the principle of the lever. For this purpose he replaces the system by " a simpler body which can nevertheless be disturbed in the same way. " Let p, j, r, s, . . . be the points of the given system to which the forces P, (), jR, S, . . . are applied. The displacement which gives the points p, g, r, s, . * . the initial virtual velocities dp, dq, dr, ds, . . . in the directions of the lines p', q\ r', s', . . . is considered. The body substituted for the system will also pass through the points p, g, r, s, . . . Similarly, its elements will have virtual velocities dp, dg, dr, ds, . . . along p', 9', r', s', . . . Fourier draws a plane perpendicular to the line p', and passing through the point p ; also a plane perpendicular to qf through the point 1 Since FOURIER'S moment is, apart from a change of sign, the modern virtual work, this conclusion is the one which is usually expressed in the form, " The virtual work of given forces is zero or negative for every displacement compatible with the constraints. " 364 THE EVOLUTION OF CLASSICAL q. These two planes intersect in the straight line d. A perpendicular, ft, is dropped from p to d. At the intersection of h and d, and in the plane perpendicular to q' passing through 5, the line hr is drawn per pendicular to d. From q the line h" is drawn perpendicular to ft'. The straight lines ft and h' are considered as the arms on an angular lever with axis d. The straight line h" is considered as a straight lever with axis d (in the plane perpendicular to q' drawn through q). Ifp is displaced along p' (by dp) the end of the arm h' is correspond ingly displaced, and the axis d can evidently be chosen in such a way that the displacement of q (required by the straight lever) is exactly equal to dq. An assembly on analogous levers can be imagined bet ween the point q and the point r, between the point r and the point s, . . . so that the system of levers thus constructed is susceptible of the same displacements as those attributed to the original system, and of this displacement alone. Fig. 107 Suppose that the forces P, (), .R, S, . . . have a total moment which is zero for the displacement dp, dg, dr, ds, . . . Because of the principle of the lever and the principle of the composition of forces, the forces P, (), jR, S, . . . will necessarily produce equilibrium in the system of levers constructed in the way that has been described. We shall show, by reductio ad absurdum, that these same forces will leave the original system in equilibrium. Indeed, if the points p, g, r, s, . . . assume the velocities dp, dg, dr, ds, and if it is assumed FOURIER 365 that the point p of the system of levers is connected with the point p of the given system, the assembly of levers will be carried along in the displacement of the given system, and the points q and g, r and r, . . . of the two systems will not separate. Therefore it can be supposed that there are connected not only the points p and p, but also the pairs of points q and <jf, r and r, s and 5, ... in the two systems. Accordingly the forces P, (), U, S, . . . will produce the motion of the two systems connected at the points p, g, r, s, . . . Now the same forces cancel each other out when applied to the system of levers alone. The reunion of the two systems could not perturb this equilibrium. Whence it is impossible that the forces P, Q9 jR, S, . . . should produce the movement of the given system. This is true for any other displacement for which the total moment of the forces is zero. " And from this can be deduced the following particular conclusion, which includes the principle of virtual velocities. If, of all the possible displacements, there is none which corresponds to a zero moment, there must be equilibrium. " Moreover it suffices that the sum of the moments should not be negative. Indeed, " it is easily proved, by the theory of the lever alone, that these forces applied [to the levers alone] cannot produce a displacement for which the total moment is positive. And since it is supposed that the presence of obstacles makes all other displacements impossible, it is necessary that when the forces act on the levers, they maintain them in equilibrium. This will still be true if the first system is applied to the second. Therefore these forces cannot separately produce the displacement in question in the first system. For this displacement would also accur if the second system were applied to the first, and we have just seen that it is then impossible. " Conversely, if some powers maintain any material system in equilibrium, there can be no displacement of the system possible for which the sum of the moments can be negative. This is proved in the following way. If it is assumed that the system can move into such a position that the moment of the forces is negative, it must be concluded that equilibrium does not exist. For the equilibrium would not cease to exist if this displacement became the only possible one. It is easy to represent this last effect by imagining assemblies of levers, similar to those which have been described above, between all the points p, 5, r, s, ... of the system, and capable of the virtual velocities which correspond to the displacement concerned. It is unnecessary to show that the equilibrium will not be disturbed by the addition of these levers. Now it is impossible that there should not be motion, because the forces will find themselves applied to an as- 366 THE EVOLUTION OF CLASSICAL sembly of levers which could not fail to be displaced if the sum of the moments of the forces were negative, just as it follows from the theory of the lever. Therefore it is necessary that the sum of the moments of the forces should never be negative. " Finally Fourier introduced the distinction between bilateral and unilateral constraints. " Whenever the displacements of which the body is capable are determined by the equations of condition which they must satisfy, the total moment of the forces cannot be positive when the forces are in equilibrium. For if this moment were positive, the moment corresponding to the contrary displacement would be negative. Now as this latter displacement is equally possible, since it satisfies the equations of condition, the forces could not cancel each other out. . . . That is why it is necessary, in this case, that the sum of the moments of the forces must be zero in order that there should be equilibrium. This is the true meaning of the principle of virtual velocities. But if the displacements are not prohibited by the equations of condition — which often happens — the equilibrium can subsist without the moment of the forces being zero, provided that it is not negative. " In connection with this demonstration, Jouguet has remarked that Fourier thus established the principle of the equivalence of cons traints, which may be stated in the following way. " Let there be a system of points acted upon by forces F and bound by constraints (L). Replace the constraints (L) by the constraints (I/) which preserve the same elementary mobility as the constraints (L). In order that the system should be in equilibrium, it is sufficient and it is necessary that the forces F should be in equilibrium under the constraints (L7). " 1 Fourier's analysis breaks down if the constraints (L) and (I/) introduce resistances to motion, even if (I/) and (L') assure the same kinematic mobility of the system. "It is not only the kinematic mobility which must be preserved but also, as it were, the dynamical mobility. " 2 1 L. AT., Vol. II, p. 171. 2 Ibid., p. 172. CHAPTER THREE THE PRINCIPLE OF LEAST CONSTRAINT (1829) The principle of least constraint was stated by Gauss in a paper called Vber ein neues Grundgesetz der Mechanik in Volume IV of the Journal de Crelle (1829).1 He wrote, " It is known that the principle of virtual velocities makes the whole of statics a matter of analysis, and that d'Alembert's principle, in its turn, reduces dynamics to statics. In the nature of things, there can exist no new principle in the science of equilibrium and motion which is not included in the two preceding principles, or which cannot be deduced from them. 44 However, such a principle may not be without value. It is always interesting and instructive to regard the laws of nature from a new and advantageous point of view, so as to solve this or that problem more simply, or to obtain a more precise pre sentation. 44 The great geometer [Lagrange] who succeeded so brilliantly in constructing mechanics from the principle of virtual velocities, had no disdain to generalise and to develop the principle of least action in Maupertuis' sense, but rather, he used it to great advantage. '* To Gauss, the principle of virtual velocities was the prototype of the principles of mechanics. But this principle was not intuitive, and demanded a special treatment in order that it might be extended from statics to dynamics. That is why Gauss believed it useful to state, in the following form, a new principle. " The motion of a system of particles connected together in any way, and whose motions are subject to arbitrary external restrictions, always takes place in the most complete agreement possible with the free motion (in moglich grosster ffbereinstimmung mil der freien Bewegung) or under the weakest possible constraint (unter moglich kleinstem Zwange). The measure of the constraint applied to the 1 The complete Works of Gauss (french edition), Vol. V, p. 25. 368 THE EVOLUTION OF CLASSICAL system at each elementary interval of time is the sum of the products of the mass of each particle with the square of its departure from the free motion. " Let m, m', m" . . . be the masses of the points of the system and a7 a , a ... be their positions at the time t. Let 6, b\ b" ... be the positions which they would assume at the time t -j- dt under the action of the forces which are applied to them, and because of their velocities at the time J, if it were supposed that they were completely free of all constraint. The actual positions, c, c', c", ... of the different points will be such that, while being compatible with the constraints, they minimise the sum The equilibrium is evidently a special case of the general law according to which m(a&)2 + ro'(a'&')2 + ™'V6")2 + • • - is a minimum. Gauss wrote, " This is how our principle can be deduced from principles already known. The force which is exerted on the particle m is evidently composed of two ; first, the force that, taking account of the velocity at the time J, brings the particle from a to c in the time dt ; secondly, the force which, in the same time, would bring the same element from c to 6, if it were supposed to be free and to start from rest. [This is the same decomposition as that which d'Alembert used.] Similarly for all particles. " By d'Alembert's principle the points m, m\ m" must be in equi librium, because of the constraints of the system, under the sole action of the second forces acting along c6, c'6', c"6", . . . According to the principle of virtual velocities this equilibrium requires that [the sum of the virtual works] should be zero for every virtual displacement which is compatible with the restraints. Or, more accurately, this sum should never be positive. " Then let y, y', y", ... be different positions of c, c', c", . . . which are compatible with the constraints. Let 0, 0', 6", ... be the angles that cy, c'y', c"y", . . . make with c6, c'b', c"6", . . . Then £mc& • cy • cos 0 will be zero or negative. THE PRINCIPLE OF LEAST CONSTRAINT 369 " Since yb* = cb2 -f- cy2 — 2ci - cy • cos 6 it is clear that y>^62 = Vm7&2 + £m~^> — 2 £mc6 - cy - cos 0. Therefore will always be positive. Therefore, finally, will always be a miniraum. Q. E. D. " In conclusion, Gauss emphasised the fact that free motions, when they are incompatible with the constraints, are modified in Nature in the same way that experimental data are modified, by the method of least squares, so as to be compatible with a necessary relation bet ween the measured quantities. 24 CHAPTER FOUR RELATIVE MOTION RETURN TO A PRINCIPLE OF CLAIRAUT CORIOLIS' THEOREMS FOUCAULT'S EXPERIMENTS 1. RETURN TO A PRINCIPLE OF CLAIRAUT (1742). The first outline of a theory of relative motion appeared, as we have seen, in Huyghens' De vi centrifugal Though he did not resolve all the difficulties of relative motion, Clairaut had the indisputable merit of generalising Huyghens' concep tions. This he did in a paper called Sur quelques principes qui donnent la solution d*un grand nombre de problemes 2 which could not have escaped the attention of Coriolis. Clairaut set out to find " what happens to any system of bodies, actuated by gravity or other accelerating forces, when this system is attached by some part to a plane and is carried with this plane in some curvilinear motion. " He introduced " the general principle for finding the motions of systems of bodies carried along by the planes on which they are placed " in the following way. " Imagine that the rectangle FGHI is placed between two curves AB and CD and that, when the corner G is moved at will on the curve AB^ the corner I follows the curve CD. " Suppose now that one of the bodies M of the system given, accelerated by gravity or by any other forces, describes the curve MJU, because of the properties of the system. " We seek the accelerating or retarding force that the motion of the plane FGHI gives the body M. We shall start by tracing the curve PQ that the point M would describe, during the motion of FGHI9 if it were fixed in the plane FGHI. We shall then determine the velo- 1 See above, Part II, p. 194. 2 Memoires de V Academic des Sciences, 1742, p. 1. RELATIVE MOTION 371 city with which it would move along this line, which will only depend on the given velocity of G and the curves AB and CD. This done, we shall seek the accelerating forces which it is necessary to suppose distributed in the space AB, CD in order that the body M, left to itself with the velocity that it has at M on Mm, might travel the line PQ Let MS, for example, represent what this force would be at M. I say that by producing MS and taking MT = MS, the straight line MT represents the force by which the motion of the plane FGHI alters the velocity of M on H f" " To prove this, I start by distinguishing the particle M from the fixed point of FGHI which corresponds to it, and I call this fixed point M'. I then remark that if, at the instant that the body M has travelled along M/* and the body M' has traveUed along M'm, the curves AB and CD were suddenly removed and the plane FGHI were allowed to move uniformly with the velocity of M' along M'm, the system which is on the plane FGHI would necessarily move in the same way as if this plane were fixed. I add to this remark that the reason why the motion along the arc Mm is altered in the curvilinear motion of the plane FGHI is that, in order to produce the curvilinear motion, it is necessary to imagine that the body M' receives an impulse MS at the instant that it has traveUed along M'm, and that the body M does 372 THE EVOLUTION OF CLASSICAL not receive this impulse. For if the body M received this impulse, the motion of the system would be exactly the same as if the plane FGHI were fixed. Given this, I say that it is the same whether M' receives an impulse and M does not, or whether M receives it in the opposite direction and M' does not. " Therefore the plane FGHI can be regarded as fixed and it can be supposed that the body experiences the action of the given forces as well as the action of the forces MT. " In short, in this way Clairaut arrives at an estimate of the quantity myr in the relative motion. This estimate is Fa — mye, where ye is the dragging acceleration. We know that this principle is incomplete. However, it led Clairaut to correct results when he confined himself to applying the principle of kinetic energy to relative motion. For it is known that Coriolis' complementary force of inertia does no work. Incomplete though it may be, Clairaut's argument has a synthetic value and, for this reason, it is of some use to complete it. This, in fact was accomplished by Joseph Bertrand in 1848.1 We shall not follow Bertrand's analysis here, but shall present an alternative method by which the argument may be completed. With respect to a fixed reference system in which the law mya = Fa is valid, the particle M describes the curve MMX between the times t and t + dt. Under the same conditions its coincident point M' is connected to a moving reference system (S) which has any arbitrary continuous motion relative to an absolute reference system. It then describes the curve M'M{ between the times t and t -f- dt. If the particle M had retained the absolute velocity va which it had at the time £, it would have travelled to M2 in the time dt, where MM2 = vadt. Simi larly, if M' had retained its absolute velocity t?e, it would have travelled 1 Journal de VScole poly technique, Vol. 32, p. 148, 1848. In this connection, J. BERTRAND brings grist to the mill of history. " Too often, after having studied analytical mechanics, a man helieves it is useless to seek to com plete this study by the reading of the scattered works with which the predecessors of LAGRANGE enriched the academic collections of the XVIIIth Century. I believe that this tendency, unfortunately very common, is such as to destroy the progress of me chanics, and that it has already produced unfortunate results. The too common custom of deducing formulae has, to some extent, led to the loss of a proper respect for the truths of mechanics considered in themselves. " In BERTRAND'S opinion, " M. Coriolis, without knowing it, has done the same as the illustrious Clairaut. " That is his opinion. It is true that CORIOLIS does not refer to CLAIRAUT. It is also true that he went further than him. But it seems unlikely to us that CLAIRAUT'S paper could have been omitted from CORIOLIS' reading. RELATIVE MOTION 373 to MS, where M'M^ = vedt. Follow on let ~vr be the relative velocity of M in the system (S) at the time t. In the triangle MM2M^ using the composition of velocities (va = ve + vr), it is seen that M^M2 = vrdt. Now give the reference system (S) and the particle M the same absolute motion defined in the following way — first, a translation jVf'Mg = _ - ye<fr2 which annuls the deviation M%M( of the point M'; secondly, the rotation — wdt at Mg, to annul the effect of the absolute rotation of the system (S), considered as a solid, between the times t and t + dt. Thus, taking account of the motion which it originally possessed, the system (S) will have experienced a rectilinear translation M'Mg of velocity ve. M2 Mjft+ctr) Fig. 109 Correspondingly, the particle M experiences the translation — ^ and, to the third order, a displacement — vdt A MiM2 - — (S A vr)dt*. These two displacements of the particle M in absolute space can be fictitiously imputed (as Clairaut realised) to forces — mye and — 2m (co A ^r) respectively. Therefore, between t and t + dt, the fundamental law of can be written in the system (S), corrected in its motion in this way (that is, in rectilinear and uniform translation with velocity vc), in the form myr == Fa — (mye + 2m (CD A vr)}. This completes and rectifies Clairaut's principle. 374 THE EVOLUTION OF CLASSICAL 2. CORIOLIS' FIRST THEOREM. CoriohV name has remained associated with the law of the com position of accelerations. This law belongs to the domain of pure kinematics — that is the way it is taught at present, before its dynamical consecpiences are explored. Historically, Coriolis was concerned with the theory of water- wheels when he embarked on his study of relative motion. This theory had already been studied by Jean Bernoulli, Euler, Borda, Navier and Ampere. To progress beyond the earlier work, it was necessary to study the following general problem. " To find the motion of any machine in which certain parts are moved in a given way. " Here we shall follow Coriolis' first paper, which was read to the Academie des Sciences on June 6th, 1831, and was printed in the Journal de r£cole polytechnique.1 Coriolis considers two reference frames. One, Ox^y^z^ is fixed —absolute. The other, OXYZ, is movable— relative. Let f, 77, £ be the absolute coordinates of the origin of the movable frame and (a, a', a"), (ft, b\ V'}, (c, c', c") be the direction cosines of the mov able axes with the fixed axes. The constraints which exist during the motion are supposed to be perfect and expressible in finite terms in the relative coordinates. Let £ be the force of constraint applied at a point of the system. Using the method of Lagrange multipliers, Coriolis writes the projections of the force on the movable axes as dL m ' dx ^ dx 2 'iL m Returning to the fixed axes, Coriolis calculates the total force — the given force and the force of constraint — acting on one point of the system. Briefly, this may be written (A) mya = P + £ Coriolis sums the equations (A) after multiplying them by the relative displacement dsr. Under these conditions the forces due to 1 21st caHer, 1832, p. 268. RELATIVE MOTION 375 the constraints vanish. But in order to perform a particular calculation, it is necessary to express the absolute accelerations ya as functions of the relative coordinates and velocities as well as the dragging motion. For this purpose Coriolis distinguishes total differentials of the true motion, indicated by the symbol d, from the differentials obtained by varying only the quantities (a, 6, ... cff). This differentiation is indicated by the symbol d^ and corresponds to a variation of the orientation of the movable system in which the quantities x, j, s, $ , 77, C remain constant. 46 If the points are not displaced relatively to the moving axes, they only have the dragging motion l corresponding to these axes, whose origin is supposed immovable and which only have a rotational motion about this origin. " Thus, " by omitting to write the denominators dt2 under the dif ferentials," Coriolis writes the components of the absolute acceleration in the form + cd*z + 2dxda + 2dydb + 2dzdc (B) d*yi = ... Terms of the equation (A) such as m then appear to him as " the components, with respect to the fixed axes, of the forces Fe which would produce the motion which each point would take if it remained in the same place with respect to the moving axes. " We have quoted from the paper of Coriolis in order to illustrate the development of his thought, how he did not pause on the kinematic aspect of the problem, but went directly to the cause that would be able to produce the dragging motion (in the modern sense now). Then Coriolis returns to the moving axes by first substituting the equations (B) in the equations (A), then by projecting on the moving axes. He obtains the relation - 2mdy(adb + o!dV + a"db") + 2mdz(adc + a'dc' + _ . „ dL . dM + ... 1 This definition does not coincide with that which is now common. 376 THE EVOLUTION OF CLASSICAL By summing for all the points of the system, after multiplying by the relative displacements dx, dy, dz, he arrives at Ydy + Zdz). The cross-terms in dx, dy, dz, vanish in the summation because of the relations between the direction cosines. If VT is the relative velocity of any point of the system with respect to the moving axes, and Pe the force which is e<jual and opposite to Fe, P the given force, Coriolis writes or, by integration, " Thus the principle of living forces is still true for motion relative to moving axes, provided that there are added to the quantities of action [that is, of work] J Pdsr cos (P-dsr) calculated from the given forces and the arcs dsr described in the relative motion, other quantities of action which are due to the forces Pe. These forces are supposed equal and opposite to those which it would be necessary to apply to each moving point in order to make it take the motion that it would have had if it were in invariably connected to the moving axes. " This is Coriolis' first theorem, which essentially belongs to the dynamics of relative motion. Coriolis applied it to the " quantity of action " transmitted to the machine which carried the movable axes. We shall not follow him in this application, where he made simultaneous use of the theorem of kinetic energy in the absolute motion and the relative motion. Coriolis remarked that when the question was that of the equilibrium of a fluid contained in a vessel turning about an axis, " it is immediately seen that it is necessary to introduce actions equal to the centrifugal forces. But it is not the same for the principle of living forces applied to the relative motion. It would be mistaken to regard this proposition as evident ; to proceed in this way for any other equation than that of the living forces would be to arrive at false results. " In conclusion, Coriolis declared, " Such are the principal results of this paper. It seems that the principle from which they stem may find many applications in the theory of machines, provided that it is supplemented by a number of propositions from rational mechanics. " RELATIVE MOTION 377 3. CORIOLIS* SECOND THEOREM. Coriolis expectation was more than fulfilled. For his first paper already contained the germ of the fundamental theorem with which his name is now associated. This vital point is his equation (C), which Coriolis did not himself analyse thoroughly, anxious as he was to cal culate the quantity of action transmitted to his water-wheels. In a second paper, Sur les equations du mouvement relatifdes systemes de corps,1 Coriolis wrote — " In this paper I give the following general proposition — that to establish an equation of the relative motion of a system of bodies or of any machine, it suffices to add to the existing forces two kinds of supplementary forces. The first are always those to which it is necessary to have regard for the equation of living forces ; that is, which are the forces opposed to those which are able to keep the part icles constantly connected with movable planes. The second are directed perpendicularly to the relative velocities and to the axes of rotation of the movable planes ; they are equal to twice the product of the angular velocity of the movable planes and the relative quantity of motion on a plane perpendicular to this axis. " The latter forces are most closely analogous to ordinary centrifugal forces. To display this analogy it suffices to remark that the centri fugal force is equal to the quantity of motion multiplied by the angular velocity of the tangent to the curve described ; that it is directed perpendicularly to the velocity and in the osculatory plane, this is, perpendicularly to the axis of rotation of the tangent. Thus in order to pass from ordinary centrifugal forces to the second forces which occur, multiplied by two, in the preceding statement, it is only necessary to replace the angular velocity of the tangent by that of the movable planes, and to substitute for the direction of the axis of rotation of this tangent, the direction of the axis of rotation of the same movable planes. In other words, it suffices to substitute everything which is related in magnitude and direction to the rotation of the tangent by what is related to the rotation of the movable planes, and to multiply the forces thus obtained by two. " It is because of this analogy that I concluded that these forces must be named compound centrifugal forces. Indeed, they have some of the characteristics of the relative motion because of the quantity of motion and some of the characteristics of the motion of the movable planes through the use of their axes of rotation and angular velocity. 1 Journal de r£cole polytechnique, 24th cahier, 1835, p. 142, 378 THE EVOLUTION OF CLASSICAL " Therefore it will be said that, for an equation of relative motion which is not that of the living forces, it is necessary to introduce twice the compound centrifugal force. " The theorem which I presented at the Academic des Sciences in 1831 consists of the disappearance of the compound centrifugal forces from the equation of the living forces. It now becomes a particular case of the more general statement on the introduction of these com pound centrifugal forces. " Coriolis9 demonstration depends directly on the equation (C) al ready written above. Indeed, if p, q, r are the " three angular velocities of the movable planes about their axes, " (C) can be written in the form dzx I d dz\ 9L dM (C) " The terms in p, q, r, dx, dy and dz in the above equation are twice the components, along the moving axes, of a force directed perpendicularly to the plane of the axis of rotation and the relative velocity. The magnitude of this force will be the product of the angular velocity Y^ p2 + g2 + r2 with the projection, on a plane perpendicular to the axis of rotation, of the quantity of motion due to the relative velocity of the particle. The sense in which this force will be carried, with respect to a motion which carries the axis of rotation towards the relative velocity, will be the same as that of the axis of rotation with respect to the velocity of rotation. " " The expressions for the forces which must be added to the given forces in order to obtain the expressions for the forces in the relative motions are — first, those which are opposed to the forces able to produce, for each particle, the motion which it would have if it were connected to movable planes ; secondly, twice the compound centrifugal forces. " This is valid for a particle of the system. Coriolis then considers the virtual velocities — the displacements dx, dy, dz in the relative mo tion — compatible with the relative constraints L = 0 M == 0 etc. These relative constraints, supposed to be perfect, will disappear on combining the equations (C), giving RELATIVE MOTION 379 .dyds - dzdy\ If ds is the actual virtual displacement and ds the virtual relative displacement ; if a, ft and y are the direction cosines of the instanta neous rotation oj(p, q, r) with respect to the moving axes and A, u, v are the direction cosines of the normal to the plane (ds, <5s), the Coriolis' complementary term becomes 2o>V m — (5s sin (rfs, ds) (od + $" + yv). Therefore, " In order to obtain an equation of the relative motion it is necessary to add to the terms ordinarily existing for absolute motion — first, those which arise from the forces which are able to force the particles to remain connected to the movable planes ; and, in addition, a term which is equal to twice the velocity of rotation multiplied by the algebraic sum of the projections, on a plane perpendicular to the axis of rotation of these planes, of all the areas of the parallelograms defined by the effective quantities of motion and the virtual velocities. " For the equation of the kinetic energy, each area is zero. For the virtual velocity coincides with the effective velocity (or rather, with the true displacement). Thus Coriolis' two theorems are linked together. We have said enough to illustrate the development of Coriolis' thought. In fact, he complicated his task by isolating the law of the composition of the accelerations — singularly hidden — from the already difficult problem of the dynamics of systems. It is rather interesting to remark in passing that Coriolis composed two acceler ations by summing , , _ at* dp ' Very fortunately, however, this procedure entailed no risk because it reduced to connecting together two terms of the unique dragging acceleration ye (in the modern sense). We stress the fact that Coriolis did not deal, in fact, with kinematics. He argued exclusively from a dynamical point of view, using forces, 380 THE EVOLUTION OF CLASSICAL and only endowed products such as mye with a physical significance. His aim was to find an equation of the relative motion which might be independent of the constraints, supposed to be holonomic and perfect. It is for this reason that he first encountered the theorem of the kinetic energy, in which the compound centrifugal force vanished. Then he was able to give a more general equation in which the complementary term appeared. All this only makes his discovery more remarkable. 4. THE EXPERIMENTS OF FoucAULT (1819-1868). In the strict sense, mechanics which is referred to terrestrial axes should take account of CoriohV compound centrifugal force. Never theless, we have already had occasion to remark l that the deviation of heavy bodies towards the East can be predicted by a very simple intuitive argument. Moreover, as early as 1833, Reich studied free fall in a mine- shaft at Freiberg (Saxony). The depth of the mine was 188 m., and he observed an average deviation of 28 millimetres in 106 separate observations. In 1851 Foucault published a paper called Demonstration physique du mouvement de rotation de la Terre au moyen du pendule.2 This demonstration made no appeal to Coriolis' work — only after the event did occur a mathematical literature. Foucault, who had been a mediocre pupil at school, was a natural physicist and an incomparable experimenter. However, he started work as the scientific member of the staff of the Journal des Debats. He set out to experiment on the direction of the plane of oscillation of a pendulum. If the observer is at first supposed at the pole (North or South) and the pendulum is reduced to a homogeneous spherical mass suspended from an absolutely fixed point, then if this point is exactly on the axis of rotation of the Earth, the plane of oscillation remains fixed in space. " The motion of the Earth, which forever rotates from west to east, will become appreciable in contrast with the fixity of the plane of oscillation, whose trace on the ground will seem to be actuated by a motion conforming to the apparent motion of the celestial sphere. And if the oscillations can continue for twenty- four hours, in this time the plane will execute a whole revolution about the vertical through the point of suspension. " But Foucault also remarked that in reality it is necessary to " take support on moving earth ; the rigid pieces to which the thread of the 1 See above, Part I, p. 63. 2 Comptes rendus de I' 'Academic des Sciences, Vol. 32, p. 135 (February 3rd, 1851). RELATIVE MOTION 381 pendulum is attached cannot be isolated from the diurnal motion. It should be borne in mind that this motion, communicated to the thread and the mass of the pendulum, might alter the direction of the plane of oscillation. " Nevertheless, experiment shows that "provided that the thread is round and homogeneous, it can be made to turn rather rapidly on itself, in one sense or another, without appreciably affecting the plane of oscillation. So that, at the pole, the experiment must succeed in all its purity. " 1 " But when our latitudes are approached, the phenomenon becomes complicated in a way that is rather difficult to appreciate. To the extent that the Equator is approached, the plane of the horizon has a more and more oblique direction with respect to the Earth. The vertical, instead of turning on itself as at the pole, describes a cone which is more and more obtuse. " From this results a slowing down in the relative motion of the plane of oscillation. This becomes zero at the Equator and changes its sense in the other hemisphere. " Without explicitly justifying the fact in his paper, Foucault assumed that the angular displacement of the plane of oscillation must be equal to the product of the angular motion of the Earth in the same time with the sine of the latitude. If the correspondence published in the collection of his works is studied in this connection, it is apparent that Foucault arrived at this relation semi-intuitively, before it had been obtained by calculations in mechanics. At first Foucault worked on a relatively modest scale by suspending a sphere of 5 kg. from a steel wire two metres long. The point of support was a strong piece of casting fixed to the top of the roof of a cellar. He took the precautions of ridding the wire of torsion and ensuring that there was no torsional oscillation of the sphere. He " encircled the sphere with a loop of organic thread whose end is attached to a point fixed on the wall, and chosen so that the oscillation of the pendulum might be 15 to 20°. " He then burnt the organic thread. This is what he observed. " The pendulum, subject to the force of gravity alone, sets off and provides a long sequence of oscillations whose plane is not slow to demonstrate an appreciable displacement. At the end of half an hour the displacement is such that it is immediately obvious. But it 1 In another place FOUCAULT reassured himself, more objectively, that " whether or not the Earth, turning, draws along the point of attachment with the monument [where the experiment was performed], the thread experiences no torsion. This implies that the hob of the pendulum submits to this motion without dragging the plane of oscillation. " 382 THE EVOLUTION OF CLASSICAL is more interesting to follow the phenomenon closely, so as to be assured of the continuity of the effect. For this purpose a vertical point, con sisting of a kind of style mounted on a support placed on the earth is fixed so that the appendicular projection of the pendulum, in its to and fro motion, grazes the fixed point when it comes to its extremitv. In less than a minute, the exact coincidence of the two points ceases. The oscillating point is continuously displaced towards the observer's left, which indicates that the deviation of the plane of oscillation takes place in the same sense as the apparent motion of the celestial sphere. . . . In our latitudes the horizontal trace of the plane of oscillation does not complete a whole circuit in twenty-four hours. " The liveliness and the accuracy of this account will be admired. As we have indicated, Foucault had started his work in a cellar. Thanks to Arago, who put at his disposal the meridian room at the Observatoire (Paris), he was later able to repeat his experiment with a pendulum 11 m. long. This provided a slower and more extensive oscillation. Finally, Foucault worked at the Pantheon (Paris) with a pendulum weighing 28 kg. suspended on a steel wire 67 m, long. As Foucault remarked — and this is an example of his remarkable intuition — " the pendulum has the advantage of accumulating the effects * and carrying them from the field of theory into that of observation. " At this point Foucault referred to a paper of Poisson,2 in which the latter has studied the deviation of projectiles. In the world of learning Foucault's experiment had the immediate success that it deserved. Notes accumulated in the Comptes rendus on the subject of the pendulum which had been revived in this way ; they included contributions from Binet, Sturm, Poncelet, Plana, Bravais, Quet, Dumas, etc. . . . Nevertheless, however brilliant it may have been, Foucault's expe riment remained rather mysterious to the general public, since it depend ed on the displacement of a plane of oscillation. Moreover, Foucault wished to give a still more tangible proof of the rotation of the Earth. The gyroscope provided him with a means of doing this. He used a pendulum suspended by its centre of gravity and executing what is called in mechanics a motion a la Poinsot. Foucault's gyroscope was a bronze fly-wheel mounted inside a metallic circle whose diameter contained a steel axis supporting the wheel. The gyroscope turned about one of its central axes of inertia, which remained fixed in space. 1 Without this accumulation FOUCAULT would not have been able to detect a force that was only the 55,000th part of the weight of the Pantheon pendulum. 2 Comptes rendus de VAcad^mie des Sciences, November 13th, 1837. RELATIVE MOTION 383 Foucault wrote1 "The body can no longer participate in the diurnal motion which actuates our sphere.2 Indeed, although because of its short length, its axis appears to preserve its original direction relatively to terrestrial objects, the use of a microscope is sufficient to establish an apparent and continuous motion which follows the motion of the celestial sphere exactly. . . . As the original direction of this axis is disposed arbitrarily in all azimuths about the vertical, the observed deviations can be, at will, given all the values contained between that of the total deviation and that of this total deviation as reduced by the sine of the latitude. " Foucault concludes, in a somewhat journalistic style that was probably natural to the reporter of the Debats — " In one fell swoop, with a deviation in the desired direction, a new proof of the rotation of the Earth is obtained; this with an instrument reduced to small dimensions, easily transportable, and which mirrors the continuous motion of the Earth itself. ... In your possession are pieces of material which are truly subject to the dragging of the diurnal motion. " Thus Fourier achieved one of Poinsot's aims. The compound centrifugal force in the sense of Coriolis, and Foucault's pendulum, are two essential achievements in mechanics ; the one has an origin which is purely mathematical, the other was the product of a physicist's brilliant intuition. Though they are united in the same rational exposition in the books that are now classical, they were born separately — it was not the reading of Coriolis that inspired Foucault's experiment. 1 Comptes rendus de VAcademie des Sciences, Vol. 35, p. 421 (September 27th, 1852). 2 More correctly, it is easy to give the gyroscope a very rapid proper rotation about its own axis, say o>, which is very large compared with the absolute rotation of the Earth., say Q. If Q is the absolute rotation of the gyroscope, Q = co + Q and the axis remains directed towards the fixed stars (U co co) as long as Q is negligible compared with co. CHAPTER FIVE POISSON'S THEOREM (1809) 1. POISSON'S THEOREM AND BRACKETS. Poisson's theorem appeared among the investigations made im mediately after the appearance of two papers by Lagrange. One of these papers appeared in 1808 and the other in 1809, and they were incorporated in the 1811 edition of the Mecanique analytique. Stimulated by the needs of the theory of perturbations in classical mechanics, they were concerned with the variation of arbitrary cons tants. Here we shall follow a paper of Poisson which was read at the Institut de France on October 16th, 1809.1 Poisson starts from Lagrange's equations Putting R = T — V, where V depends only on the ql and not on the y », he obtains dR dT 5-7 = 5-7 = ut dqi dql whence du1_dR W dt ~ d^ 66 In this way the equations of motion are reduced to the simplest form that they can be given. " 1 Journal de MZcole polytechnique, cahier XV, 1809, p. 266. While following the development of POISSON'S analysis rigorously, we have taken the liberty of condensing its form by using the convention of the summation of dummy suffixes. This is commonly used in the absolute differential calculus and allows the direct consideration of a system of k degrees of freedom (rather than three, as POISSON did). Further, we have introduced the distinction between the symbols of partial and total derivatives. POISSON 385 The new variables ut are functions of the q,, and the gj. Con versely, the q'i can be regarded as functions of the ql and the ut. Considering R as a function of the qt and the ut, Poisson denotes the partial derivatives of jR when the independent variables are ql and 8R Ui by - — ; and the derivatives of J? when the independent variables vqi , df?\ are q^ and q't by ( — ). Thus equation (1) becomes dut _ /dR ~dt"~ \dq However ( ' dqt dq so that dui BR Tke partial derivatives of R may be written Finally, by calculating - - in two different ways, it is found that OUj, OUj 2 = Uk ^ | w dUiduj dul 92q'k , Bqi - - -- J -- =- . From this is obtained the relation which will be used in the sequel. Given this, Poisson considers a first integral of the equations of motion containing a single arbitrary constant a. This integral equation, if solved for a, would lead to a = funct (qi ... q^ Ui . . . u&, t). 25 386 THE EVOLUTION OF CLASSICAL Hence . 3a , da , . da , 0 = ar da da , da fdR _ dq'r\ ^kqk+^k \dqk~~ Ur 3qJ ' Differentiating with respect to q^ «rt n _ L i^\ + ^ ^ + -^ W dt \~dqj "*" Sqk dqt "*" 8uk If another first integral of the equations of motion, containing an arbitrary constant 6, had been considered, then m o - A ffl 4- 1* ^ + ^- f 82jR - « l/ j <ft \ag/ "^ a9& dq> ^ duk T By multiplying equation (6) by — and equation (7) by — — , sum- OUt OUi ming over the dummy suffixes and adding the two equations, 0 - — ™ (—] — —— (—} 4-^(— — — ——} ^ ' "" dui dt \dqi/ dut dt \dqi/ dqi \dqk dui dqk duj' Differentiating the equation from which (6) was obtained with respect to ut, instead of q^ there is obtained _ u r dt \3tti / dqk 3ut duk \dqkdui dqkdqi/ duk dqk The third term vanishes because of (4). There remains (£tt\ t\ d 1 3^\ ^a dqic da dqi dt \duj dqk dui duk dqk' Also, for the other first integral, dt \diii/ dqk dui duk dqk Multiplying (6") by — — and (7") by — , summing over the dummy suffixes and adding the two equations, .---- - dt \du dqi dqi dt \du dqL idb_ 9a _ db_ 9a \ dqi idb_ da_ _ jtt da_ \ dui \dqk dqi dqi dqj dqk \duk dqi dqi duk/ ' POISSON 387 The third term of this equation is zero because of the relation (5). If the suffixes i and k are interchanged in the fourth term, by adding (8) to (8') it follows that y d (da\ da d^ fdb\ da^ d_ /db\ __ db_ d_ ida\ _ ~i di (dqj ~"faidt \dq~J + dq'i dt \duj dqi dt \duj ~~ db d du or d [ db da da db \ A _ _ — _ — = u dt or finally db da db da — - -- — — = constant dui dqi dqt dut or (fe9 a) = constant where (i, a) is an expression which has become known as a " Poisson bracket. " It is evident that (6, a) = — (a, b) and that (a, a) = 0. Poisson concludes, " The analysis that we have just performed therefore leads us to this remarkable result — that if the values of the arbitrary constants on the integrals of the equations of motion of a system of bodies are expressed as functions of the independent variables (q,) and the quantities (ut), the combination of the partial differentials of these functions that is represented by (a, b) will always be a constant quantity. " This proposition, which has become classical, evidently exhibits considerable aesthetic value. Its practical content is more limited. Indeed, Poisson's theorem seems to indicate that it is sufficient to know two first integrals of the equations of motion in order to be able to deduce a third from them ; by combining this with one of the first two, a fourth would be obtained, etc ---- But if the bracket (a, b) is identically constant, or if it is a function of the integrals already known, this process contains nothing new. 2. THE LAGRANGE-POISSON SQUARE BRACKETS. Poisson supposes that, to the right hand side of the Lagrange equa tions dt \Sq' 388 THE EVOLUTION OF CLASSICAL is added a term depending on the function of the perturbing forces, Q. This yields Since the variables u£ are always defined by — , Lagrange's equa- tions take the form The expression I — ) must always be interpreted as a derivative in which the qi and the q[ are chosen as the independent variables. This distinction does not apply to derivatives of £?, which is supposed to be independent of the g£. If the equations (1) are integrated completely, so that the solution contains 2k arbitrary constants a,, it is desired to satisfy the equations (2) by varying the arbitrary constants. Since the number of these is twice the number of the equations (2), the 2k quantities as can be restricted by any k conditions that may be chosen. Poisson supposed, " as in the theory of the Planets " that the dif ferentials of the variables gf kept the same form independently of whether the as were constants or not. He goes on to express this condition by the k relations tA\ * d^ i A (£ = 1, 2 . .. k) (4) dqt = ^- das = 0 } / 3 das (s from 1 to 2k). On the other hand ^ dt + ™ *. But when the a, are constants, the first two terms are equal because of (1). Accordingly, if only the as vary (5) du^^-dt. oqi T, j| The 2k equations (4) and (5), which are linear and of the first order in the quantities da,, determine the differentials of the arbitrary cons tants. Let ar = funct (t, 31, ... qk, MI, ... uk) be a first integral of the equations (1). Then put , dar dQ _ dar = — - -— dt . POISSON 389 Now X*_ = ?R.fa. dqj das dqj Therefore , dar das dQ _ /j from 1 to k \ ^ * T duj dqj das \s from 1 to 2k' But in the derivatives — are necessarily zero. Therefore 3Uj 0 = — = — ^i duj 3as duj or, by slimming the equations (7) after multiplying them by — -, dqj (8) Q==3Qda.dar das duj dqj' By subtracting (8) from (6) 7 dQ . (dar das das dar\ and, using the definition of the Poisson brackets, , SQ . . . fr = 1, 2 . . . 2Jfc\ Jar == — - dt (ar, as) ( I . das \s from 1 to 2k I The brackets (ar, as) are functions only of the arbitrary constants a-L, . . . . , a2fc. " It follows that, in the equations of mechanics, the first differentials of the arbitrary constants can be expressed by means of the partial differences of the function Q, taken with respect to these quantities and multiplied by functions of these same quantities, which do not contain the time explicitly. This is the beautiful theorem that Mr. Lagrange and Mr. Laplace first discovered in connection with the differences of elliptic elements, and which Mr. Lagrange then extended to a system of any bodies subject to forces directed towards fixed or movable centres and whose intensities are functions of the distances of the bodies from these centres. " Lagrange had arrived at the formulae -—dt= [ar, aj das dar in which the square-bracket expression had the value !-„ ^i_?2i^_M^ Lar'asJ~aar das BasdOr' CHAPTER SIX ANALYTICAL DYNAMICS IN THE SENSE OF HAMILTON AND JACOBI 1. HAMILTON OPTICS. ITS DOUBLE INTERPRETATION IN TERMS OF EMISSION AND WAVE PROPAGATION. Hamilton's ideas on dynamics cannot be divorced from his ideas on optics. For this reason it is essential that we should, for a few moments, concern ourselves with the latter. We shall follow the edition of Hamilton's works that has been published by the Royal Academy of Ireland. Apart from the papers which have been known and classical for some time, this edition, very fortunately, contains extracts from the numerous note-books which Hamilton kept and which had not been published before. No doubt the author, considering them minor works, had not wished to make them public — but they throw the work of this inspired Irishman into a new and very interesting light.1 In the first place we shall cite an article which appeared in the Dublin University Review 2 for October, 1833, called On a general method of expressing the paths of light, and of the planets, by the coeffi cients of a characteristic function. At the time that Hamilton started his investigations in optics, neither the theory of waves nor the emission theory were generally accepted. Hamilton's geometric optics, which was essentially a new method of formalising the coUection of results that had already been obtained, was capable of being interpreted in terms of wave propagation (in Huyghens* sense) and corpuscles (in the sense of the dynamical principle of least action). This was the essential merit of his theory, 1 The Mathematical Papers of Sir William Rowan Hamilton. Vol. I, Geometrical Optics edited for the Royal Irish Academy by A. W. CONWAY and J. L. SYNGE (1931) ; Vol. II, Dynamics edited for the R. I. A. by A. W. CONWAY and J. McCoNNEL (1940). Cambridge University Press. 2 Pp. 795-826. HAMILTON AND JACOBI 391 which, can only seem more meritorious to our modern age in which a similar dualism has been established in theoretical physics. A great admirer of Lagrange, Hamilton declared, in the article referred to above, that he was " struck by the imperfection of deductive mathematical optics. " He wished to give to optics, on the plane of formal theory, the same " beauty, power and harmony " with which Lagrange had been able to endow mechanics. I repeat that it was certainly the formalism which concerned him. " Whether we adopt the Newtonian or the Huyghenian, or any other physical theory, for the explanation of the laws that regulate the lines of luminous or visual communication, we may regard these laws themselves, and the properties and relations of these linear paths of light, as an important separate study, and as constituting a separate science, called often mathematical optics. " l Hamilton recalled the development which we have already studied, Fermat, Maupertuis, Eider, Lagrange. w But although the law of least action has thus attained a rank among the highest theorems of physics, yet its pretensions to a cosmological necessity, on the ground of economy in the universe, are now generally rejected. And the rejection appears just, for this, among other reasons, that the quantity pretended to be economised is in fact often lavishly expended. " This, for instance, is what is shown in the commonplace case of reflexion on a spherical mirror, where obviously if one of the rays issuing from a point is minimal, the other corresponds in fact to a maximal. We can therefore speak reasonably only of a stationary property of the action (or an extremal one, as understood in the calculus of variations). " We cannot, therefore, suppose the economy of this quantity to have been designed in the divine idea of the universe : though a simplic ity of some high kind may be believed to be included in that idea **. Such are the rational motives which led Hamilton, at the same time as he retained the consecrated term action, to speak, in optics as in dynamics, of stationary or varying action, according to whether the extremities of the rays or trajectories are fixed by hypothesis or not. I shall pass over the remarkable statement of the principles of the calculus of variations contained in the paper we are analysing, and come 1 Incidentally HAMILTON did not hesitate to state his doctrine of scientific philo sophy. Thus he distinguished a stage in which the facts are raised to laws by induction and analysis, and another in which the laws are used to obtain consequences by deduction and synthesis. This was formulated in the following remarkable passage. ** We must gather and group appearances, until the scientific imagination discerns their hidden laws, and unity arises from variety ; and then from unity we must rede- duce variety, and force the discovered law to utter its revelations of the future. " Better than a fine formula, this thesis is the expression of the method of work that HAMILTON always followed. 392 THE EVOLUTION OF CLASSICAL to the exact statement of the Hamiltonian principle of stationary action in optics : " The optical quantity called action, for any luminous path having i points of sudden bending by reflexion or refraction, and having therefore i + 1 separate branches, is the sum of i + 1 separate integrals, ACTION = V = vjrfl/i = Fi + F2 + . . . + Vl+l of ichich each is determined by an equation of the form the coefficient vt of the element of the path, in the ith medium, depending, in the most general case, on the optical properties of that medium, and on the position, direction and colour of the element, according to rules discovered by experience. (For example, if the ith medium is an ordinary medium, vt is its refractive index.) This quantity V is stationary in the propagation of the light. " The law of varying action is a generalisation of the stationary law in which the ends of an optical (luminous) path are allowed to vary. The conditions at the limits thus make necessary the intervention of finite difference equations of the type A V = Xu = 0 on each surface u = 0 of reflection or refraction. In passing, Hamilton indicated that the remarkable permanence of what he called the components of normal slowness (inversely proportional to those of the velocity of wave propagation in Huyghens' sense) had been suggested to him by the observation that the characteristic function V is such that the wave surfaces satisfy the equation V = Constant. The components of normal slowness are nothing else than the partial , . . dV dV dV derivatives — , — * -5-- Thus is rediscovered the theorem — then ox oy oz disputed — of Huyghens according to which the rays of every homo geneous system, starting from a single point or normal to a surface, remain normal to a family of surfaces after they have been subjected to any number of reflections or refractions. We learn a little more about Hamilton's procedures in optics by following the Third Supplement to an Essay on the Theory of Systems of Rays. First, calling the initial and final coordinates of a ray (x',yr, zf) and (x, y, z), Hamilton writes <A) *-+» + »-*--« HAMILTON AND JACOBI 393 (where a, /?, y and a', /?', y' are the direction cosines of the ray at its end and beginning respectively) or On the other hand, the conditions for an extremum require that w E*-*E-- E*-'S <* The function t; is homogeneous in a, /? and 7 and also depends on the frequency of the light. Hamilton expresses the latter fact by the introduction of a chromatic index %. Thus he arrives at the following two equations in the first partial derivatives SV 3V . /-SV —dV -SV (C) The similarity of the form of these equations with that of the equations of dynamics is evident — V corresponds to the action integral (in the Euler-Lagrange sense) ; the equation (C) corresponds to the equation of kinetic energy and % to a certain function of the total energy. Moreover, Hamilton's optical equations can be easily written in the canonical form that he himself gave to the equations of dynamics. It is sufficient to denote the components of normal slowness (or the partial derivatives of V) by cy, r and v to write dx ___ 9flj da __ dQ • 7 Tjr ~^ 1 • • • ~7~T> ^ etC. dV do* dV dx As we have already indicated, Hamilton interpreted the action F, in the language of the wave theory, as the time necessary for a wave of frequency % which starts from the point (x' ', y', 2') to travel to the point (#, y, z). If the wave velocity (ondulatory velocity) of propagation along the corresponding radius is called u, the relation (D) u - 1 v / v or, more generally, 394 THE EVOLUTION OF CLASSICAL allows V to be written as F=J"V" u (g g * y , y Since the rays are identical in the two theories, to the extrema dV= dfvds = 0 of the ^mission theory there corresponds the extremal which is Fermat's principle. There is, here, the germ of the transcription which Schrodinger was to turn to good use in dynamics, in generalising equation (D) by the introduction of a group velocity (in Rayleigh's sense) identified with v. 2. THE DYNAMICAL LAW OP VARYING ACTION IN HAMILTON'S SENSE. Historically, Hamilton's first work in dynamics is contained in a manuscript dated 1833 and called The Problem of Three Bodies by my Characteristic Function.3- He treated the problem of the Sun, Jupiter and Saturn and introduced, from the beginning, the charac teristic function. V= f 2Tdt. Jo (The living force accumulated from the origin of time to the time t.) Hamilton showed that this function must satisfy two equations in the partial derivatives of the first order. He then compared an approx imate solution of this problem with that obtained by Laplace, studied the perturbations, determined the characteristic function of elliptical motion and established the equation 3V — =zz t (A, constant of living forces) . He then proved that the two equations connecting the partial derivatives of V have a common solution, and directed his attention to the deter mination of this solution by successive approximations. Therefore this paper already contained essential results. We shall not, however, further discuss it, for Hamilton undertook the codification of these investigations in two fundamental papers which were published in 1834. We propose to analyse these. 1 Note-book 29. HAMILTON AND JACOBI 395 In his statement of the intentions of his First Essay on a General Method in Dynamics l Hamilton recalled that the determination of the motion of a system of free particles, subject only to their mutual attraction or repulsion, depended on the integration of a system of 3 (n — 1) ordinary differential e<juations of the second order or, by a transformation due to Lagrange, on a system of 6(n — 1) ordinary differential equations of the first order. Hamilton reduced this problem to the " search and differentiation of a single function " which satisfied two equations of the first order in the partial derivatives. From this transference of the difficulties, even if it is thought that no practical advantage results, " an intellectual pleasure may result from the reduction of the most complex and, probably, of all researches respecting the forces and motions of bodies, to the study of one characteristic function, the unfolding of one central relation. ..." And Hamilton adds, " this dynamical principle is only another form of that idea which has already been applied to optics in the Theory of systems of rays. . . . " Starting from the classical equation (1) £ m (*"<5* + y% + ****) = <5U" with 17 = £ mm'/(r), in which U is the function of forces, Hamilton denotes the living force of the system by 2T = E ™ (*'2 + /2 + *'2)> and writes the law of living forces in the form T = U + H. The quantity H — which it has become customary to call the Hamil- tonian of the system — is independent of the time in a given motion of the system. But when the initial conditions are varied, H varies correspondingly according to the equation dT = SU + 6H. On multiplying by df, integrating from 0 to f, using equation (1) and the equation that defines the kinetic energy, Hamilton obtains J^ 2 m (dxM + dydy' + <W) = J"o £ m (dx'dx + dy'dy + dz'dz) + Then, by means of the calculus of variations (A) 6 V= £ m (x'dx + y'dy + z'dz) — £ m (a'da + b'db + c'8c) + tdH 1 Phil Trans. Roy. Soc. (1834), II, p. 247. 396 THE EVOLUTION OF CLASSICAL where (#, y, z) and (a, &, c) are the final and initial coordinates of the points of the system, and V is the function (B) v = Jo S m (x'dx + y'Jy +